Indefinite integrals (part II)

WEBVTT
Kind: captions
Language: en

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Welcome back.

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In this presentation I'm just
going to do a bunch of examples

00:00:04.410 --> 00:00:08.200
of taking the antiderivative or
the indefinite integral of

00:00:08.200 --> 00:00:11.610
polynomial expressions, and
hopefully I'll show you that

00:00:11.610 --> 00:00:13.740
it's a pretty straightforward
thing to do.

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So let's get started.

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If I wanted to take indefinite
integral-- and you could do a

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web search for integral and
you'll see this drawn

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properly-- take the indefinite
integral-- let me make

00:00:29.190 --> 00:00:31.060
a big expression.

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Let's say I want to take the
indefinite integral of 3x to

00:00:37.930 --> 00:00:49.650
the negative 5 minus 7x
to the third plus 3

00:00:49.650 --> 00:00:54.250
minus x to the ninth.

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So you might already
be intimidated by

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what I wrote down.

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Well, one, if you saw the
last presentation or if you

00:01:01.700 --> 00:01:03.035
understood presentation, you
probably realize, well the

00:01:03.035 --> 00:01:04.690
indefinite integral even
though it looks like fancy

00:01:04.690 --> 00:01:06.770
math isn't that fancy.

00:01:06.770 --> 00:01:10.020
Or at least it isn't that
difficult to perform.

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And all you have to realize now
is if we took the derivative of

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a polynomial, it was just the
sum of the derivatives

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of each of the terms.

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it actually it turns out is the
same way the other way around.

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The antiderivative of this
entire expression is just the

00:01:27.990 --> 00:01:32.160
sum of the antidervatives of
each of the individual terms.

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So we can just take
the [? integers ?]

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of each term and we'll
get the answer.

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So what does this equal?

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Well in this case 3x
to the minus 5 power.

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So we take the exponent, we add
1 to the exponent, so now we

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get x to the negative 4, and
then we multiply the

00:01:52.470 --> 00:01:58.380
coefficient times 1
over the new exponent.

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So 1 over the new
exponent is minus 1/4.

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So 3 times minus
1/4 is minus 3/4.

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And let's see.

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Here we have x to the third.

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So instead of x to the third,
let's raise it by one number.

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So we get x to the fourth.

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And then we multiply
the coefficient.

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You know, we could either just
keep the minus and say the

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coefficient's 7, or we could
just say the coefficient

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is minus 7.

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We multiply the coefficient
times 1 over the new exponent.

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So the new exponent is 4,
so we multiply 1/4 times

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minus 7, so minus 7/4.

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And now this is interesting.

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3, just 3.

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Well how do we apply this?

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Well isn't 3 the same thing
as 3 times x to the 0?

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Right, because x to
the 0 is just 1.

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And that's how you
should view it.

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It shows you that this rule
is actually very consistent.

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So what's the answer
derivative of 3?

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Well if we view 3 as 3 x to the
0, we raise the exponent by

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1, so now we're going
to have x to the 1.

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And x to the 1 is just
x, so I'm just going

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to leave it as an x.

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And we multiply it, the old
coefficient-- this 3 or you

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know the derivative
coefficient-- we multiply that

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times 1 over the inverse
of the new exponent.

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So the exponent's 1, so
the inverse of 1 is 1,

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so it just stays 3.

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We've multiplied 3 times
1/1, which is still just 3.

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And then finally x to the
ninth-- I think you're getting

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the hang of this-- we raise the
exponent by one,

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x to the tenth.

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And then we multiply the
current coefficient.

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Well the current coefficient
is minus 1, right.

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We just didn't
write the 1 there.

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We multiply the current
coefficient minus 1 times

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1 over the new exponent,
so it's minus 1/10.

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There we did it.

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That wasn't too difficult of
taking the antiderivative

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or-- I always forget.

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Plus c, right?

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Because when you take the
derivative of any constant

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it becomes 0, so it might
have disappeared here.

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So plus c where this
is any constant.

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This could be a 10, could
be a million, could

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be a minus trillion.

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It's any constant.

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And just to really hit the
point home, let's take the

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derivative of this and just
make sure we got

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this expression.

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And hopefully this is second
nature to you by now.

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And you know if you ever run
out of practice problems

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in your book because you
love doing math so much,

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just make up problems.

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That's what I'm doing.

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I do this even when I'm not
recording videos, just for fun.

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So let's take the
derivative of this.

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Minus 4 times this coefficient.

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Minus 4 times minus 3/4 is 3x.

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Then we subtract 1 from
this exponent, minus 5.

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And then 4 times 4 is minus 7 x
to the-- we take 1 from this

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exponent-- x to the third.

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And I promise you I'm not
even looking up here.

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I know you might think, well
Sal, he's just looking up here,

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but no I'm actually in my head
at least working through this.

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And then plus the
derivative of 3x.

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Well the derivative of 3x is
3-- is almost second nature

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now, but you can kind of
do this-- is 3x to the 1.

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And you say 1 times 3
is 3 times x to the 0.

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And then 10 times minus 1/10.

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Well that's just minus 1.

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x to the 1 less than 10, so x
to the ninth, plus-- what's the

00:05:43.670 --> 00:05:45.690
derivative of any constant?

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Right, it's 0.

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You could almost do this
constant as some number

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times x to the 0.

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And if you took the derivative,
well you multiply the 0

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times c and you get 0.

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Well, you might get minus
1 depending on how

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you're doing it.

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But that's actually kind of
an interesting question.

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OK I'll stop digressing.

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But you get a 0 here, and if
you simplify that, that just

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equals 3x to the minus 5
minus 7x to the third plus

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3 minus x to the ninth.

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Think we have time for one
more problem like this.

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I think you probably got this.

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This is probably one of the
more straightforward things

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you'll learn in mathematics.

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And in future presentations
I'll give you more of an

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intuition of why the
antiderivative is useful.

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We're learning the indefinite
integral, but we could learn to

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use the definite integral,
which we'll learn in a couple

00:06:35.045 --> 00:06:38.500
of presentations to figure out
things like the area under

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curve, or the volume of
a rotational figure.

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Well I don't confuse
you too much.

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Let's do one more problem.

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I won't make this one as hairy.

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So the integral of negative
1/2x to the minus 3

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plus 7x to the fifth.

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Let's start with this
term of the polynomial.

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We raise the exponent one,
so x to the minus 2 now,

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right, because we added
one to negative 3.

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And then we multiply 1 over
this new exponent times

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the old coefficient.

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And actually I'll write
out all the steps.

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So the old coefficient
is minus 1/2.

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So this is a minus 2.

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Minus 2 so we multiply
it times minus 1/2.

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Let me switch colors back.

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Plus we raise the exponent by
one, x to the sixth, and we

00:07:49.410 --> 00:07:54.140
multiply the old coefficient
times 1 over the new

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coefficient, times 1/6.

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And so what's the answer?

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Well what's minus 1/2
times minus 1/2?

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Well that's positive
1/4 x to the minus 2.

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Oh, and of course, plus c.

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As you can tell, this is
my main source of missing

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points on calculus quizzes.

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1/4 x to the minus 2 plus
7/6 x to the sixth plus c.

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There you go.

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And if you wanted to take the
derivative, minus 2 times 1/4

00:08:35.160 --> 00:08:40.420
is minus 2/4 which is minus
1/2 x to the minus 3.

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And then 6 times 7/6 is 7x.

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And then you decrease
the exponent by one,

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x to the fifth.

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And the derivative of
our constant is 0.

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And then we get our
original expression.

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Hopefully at this point you're
pretty comfortable taking a

00:08:57.000 --> 00:09:00.540
derivative of a polynomial, and
then given a polynomial you can

00:09:00.540 --> 00:09:02.500
actually take the
antiderivative, go

00:09:02.500 --> 00:09:03.980
the other way.

00:09:03.980 --> 00:09:06.770
And never forget to
do your plus c.

00:09:06.770 --> 00:09:08.500
And I hope you understand why
we have to put that constant

00:09:08.500 --> 00:09:14.460
there, because when you take an
antiderivative, you don't know

00:09:14.460 --> 00:09:17.350
whether the original thing that
you the derivative of I guess

00:09:17.350 --> 00:09:20.600
had a constant there, because
the constant's derivative is 0.

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Hopefully I confused you
with that last statement.

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I'll see you in the next
presentation and I'll show you

00:09:24.320 --> 00:09:26.450
how to reverse the chain rule.

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See you soon.


The Indefinite Integral or Anti-derivative

WEBVTT
Kind: captions
Language: en

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Welcome to the presentation
on the indefinite integral

00:00:03.210 --> 00:00:04.430
or the antiderivative.

00:00:04.430 --> 00:00:06.750
So let's begin with a
bit of a review of the

00:00:06.750 --> 00:00:07.270
actual derivative.

00:00:07.270 --> 00:00:10.700
So if I were to take
the derivative d/dx.

00:00:10.700 --> 00:00:13.450
It's just the
derivative operator.

00:00:13.450 --> 00:00:16.700
If I were to take the
derivative of the expression

00:00:16.700 --> 00:00:20.140
x squared-- this is an easy
one if you remember the

00:00:20.140 --> 00:00:21.860
derivative presentation.

00:00:21.860 --> 00:00:23.400
Well, this is pretty
straightforward.

00:00:23.400 --> 00:00:24.640
You just take the exponent.

00:00:24.640 --> 00:00:27.100
That becomes the new
coefficient, right.

00:00:27.100 --> 00:00:29.040
You actually multiply it times
the old coefficient, but in

00:00:29.040 --> 00:00:32.310
this case the old coefficient
is 1, so 2 times 1 is 2.

00:00:32.310 --> 00:00:35.130
And you take the variable 2x.

00:00:35.130 --> 00:00:37.170
And then the new exponent
will be one less than

00:00:37.170 --> 00:00:38.460
the old exponent.

00:00:38.460 --> 00:00:41.930
So it'll be 2x to
the 1, or just 2x.

00:00:41.930 --> 00:00:42.580
So that was easy.

00:00:42.580 --> 00:00:46.000
If I had y equals x squared we
now know that the slope at any

00:00:46.000 --> 00:00:50.240
point on that curve,
it would be 2x.

00:00:50.240 --> 00:00:52.040
So what if we wanted
to go the other way?

00:00:52.040 --> 00:00:55.720
Let's say if we wanted to start
with 2x, and I wanted to say

00:00:55.720 --> 00:01:07.330
2x is the derivative of what.

00:01:07.330 --> 00:01:09.420
Well, we know the answer
this question, right?

00:01:09.420 --> 00:01:10.755
Because we just took the
derivative of x squared

00:01:10.755 --> 00:01:12.170
and we figured out 2x.

00:01:12.170 --> 00:01:14.680
But let's say we didn't
know this already.

00:01:14.680 --> 00:01:18.170
You could probably figure it
out intuitively, how you can

00:01:18.170 --> 00:01:21.070
kind of do this operation
that we did here, how

00:01:21.070 --> 00:01:23.450
you can do it backwards.

00:01:23.450 --> 00:01:27.830
So in this case the notation--
well we know it's x squared--

00:01:27.830 --> 00:01:31.870
but the notation for trying to
figure out 2x is the derivative

00:01:31.870 --> 00:01:35.920
of what, we could say that--
let's say 2x is the

00:01:35.920 --> 00:01:39.410
derivative of y.

00:01:39.410 --> 00:01:43.050
So 2x is the derivative of y.

00:01:43.050 --> 00:01:46.150
Let's get rid of this of what.

00:01:46.150 --> 00:01:47.270
Then we can say this.

00:01:47.270 --> 00:01:51.260
We can say that y is equal to--
and I'm going to throw some

00:01:51.260 --> 00:01:55.990
very fancy notation at you and
actually I'll explain why we

00:01:55.990 --> 00:01:59.500
use this notation in a couple
presentations down the road.

00:01:59.500 --> 00:02:01.680
But you just have to know at
this point what the notation

00:02:01.680 --> 00:02:03.925
means or what it tells you to
really do, which really is

00:02:03.925 --> 00:02:06.070
just the antiderivative or
the indefinite integral.

00:02:06.070 --> 00:02:10.340
So we could say that y is
equal to the indefinite

00:02:10.340 --> 00:02:14.350
integral 2x dx.

00:02:14.350 --> 00:02:17.220
And I'm going to explain what
this squiggly line here is and

00:02:17.220 --> 00:02:20.690
dx, but all you have to know is
when you see the squiggly line

00:02:20.690 --> 00:02:24.700
and this dx and then something
in between, all they're asking

00:02:24.700 --> 00:02:28.370
is they want you to figure out
what the antiderivative

00:02:28.370 --> 00:02:30.060
of this expression is.

00:02:30.060 --> 00:02:32.550
And I'll explain later
why this is called the

00:02:32.550 --> 00:02:33.340
indefinite integral.

00:02:33.340 --> 00:02:36.350
And actually this notation
will make a lot more sense

00:02:36.350 --> 00:02:39.970
when I show you what a
definite integral is.

00:02:39.970 --> 00:02:42.000
But let's just take it for
granted right now that an

00:02:42.000 --> 00:02:44.000
indefinite integral-- which I
just drew here, it's kind of

00:02:44.000 --> 00:02:47.450
like a little squirrely thing--
is just the antiderivative.

00:02:47.450 --> 00:02:52.350
So y is equal to the
antiderivative essentially,

00:02:52.350 --> 00:02:56.150
or the indefinite integral
of the expression 2x.

00:02:56.150 --> 00:02:57.270
So what is y equal to?

00:02:57.270 --> 00:03:02.210
Well y is obviously
equal to x squared.

00:03:02.210 --> 00:03:03.220
Let me ask you a question.

00:03:03.220 --> 00:03:06.830
Is y just equal to x squared?

00:03:06.830 --> 00:03:08.660
Because we took the derivative,
and clearly the derivative

00:03:08.660 --> 00:03:10.575
of x squared is 2x.

00:03:10.575 --> 00:03:14.320
But what's the derivative
of x squared-- what's the

00:03:14.320 --> 00:03:15.880
derivative x squared plus 1?

00:03:21.090 --> 00:03:24.500
Well, the derivative of
x squared is still 2x.

00:03:24.500 --> 00:03:26.100
What's the derivative of 1?

00:03:26.100 --> 00:03:28.460
Right, derivative of 1
is 0, so it's 2x plus

00:03:28.460 --> 00:03:30.540
0, or still just 2x.

00:03:30.540 --> 00:03:37.570
Similarly, what's the
derivative of x squared plus 2?

00:03:37.570 --> 00:03:39.050
Well the derivative of
x squared plus 2 once

00:03:39.050 --> 00:03:42.620
again is 2x plus 0.

00:03:42.620 --> 00:03:45.200
So notice the derivative
of x squared plus

00:03:45.200 --> 00:03:47.890
any constant is 2x.

00:03:47.890 --> 00:03:52.390
So really y could be x
squared plus any constant.

00:03:52.390 --> 00:03:55.420
And for any constant
we put a big c there.

00:03:55.420 --> 00:03:56.960
So x squared plus c.

00:03:56.960 --> 00:03:59.100
And you'll meet many calculus
teachers that will mark this

00:03:59.100 --> 00:04:01.600
problem wrong if you forget to
put the plus c when you do

00:04:01.600 --> 00:04:03.340
an indefinite integral.

00:04:03.340 --> 00:04:07.360
So you're saying Sal, OK,
you've showed me some notation,

00:04:07.360 --> 00:04:10.880
you've reminded me that the
derivative of any constant

00:04:10.880 --> 00:04:14.640
number is 0, but this really
doesn't help you solve

00:04:14.640 --> 00:04:15.270
an indefinite integral.

00:04:15.270 --> 00:04:18.950
Well let's think about a way--
a systematic way if I didn't do

00:04:18.950 --> 00:04:21.200
it for you already--
that we could solve an

00:04:21.200 --> 00:04:23.480
indefinite integral.

00:04:23.480 --> 00:04:24.540
Let me clear this.

00:04:30.440 --> 00:04:33.615
A bolder color I think would
make this more interesting.

00:04:36.300 --> 00:04:45.300
Let's say we said y is equal to
the indefinite integral of--

00:04:45.300 --> 00:04:47.220
let me throw something
interesting in there.

00:04:47.220 --> 00:04:54.350
Let's say the indefinite
integral of x cubed dx.

00:04:54.350 --> 00:04:58.510
So we want to figure out some
function whose derivative

00:04:58.510 --> 00:05:01.470
is x to the third.

00:05:01.470 --> 00:05:02.620
Well how can we
figure that out?

00:05:02.620 --> 00:05:05.540
Well just from your intuition,
you probably think, well it's

00:05:05.540 --> 00:05:10.420
probably something times x
to the something, right?

00:05:10.420 --> 00:05:19.116
So let's say that y is
equal to a x to the n.

00:05:19.116 --> 00:05:27.910
So then what is dy/dx, or
the derivative of y is n.

00:05:27.910 --> 00:05:29.390
Well we learned this in
the derivative module.

00:05:29.390 --> 00:05:32.320
You take the exponent, multiply
it by the coefficient.

00:05:32.320 --> 00:05:34.480
So it's a times n.

00:05:37.890 --> 00:05:42.820
And then it's x to
the n minus 1.

00:05:42.820 --> 00:05:46.810
Well in this situation we're
saying that x to the third is

00:05:46.810 --> 00:05:50.330
this expression, it's
the derivative of y.

00:05:50.330 --> 00:05:52.500
This is equal to
x to the third.

00:05:52.500 --> 00:05:58.220
So if this is equal to x to
third, what's a and what's n.

00:05:58.220 --> 00:06:00.360
Well, n is easy to figure out.

00:06:00.360 --> 00:06:02.670
n minus 1 is equal to 3.

00:06:02.670 --> 00:06:07.430
So that means that
n is equal to 4.

00:06:07.430 --> 00:06:10.190
And then what is a equal to?

00:06:10.190 --> 00:06:14.770
Well a times n is equal to 1,
right, because we just have a 1

00:06:14.770 --> 00:06:18.410
in this coefficient, this has
a starting coefficient of 1.

00:06:18.410 --> 00:06:20.255
So a times n is 1.

00:06:20.255 --> 00:06:23.210
If n is 4, than a must be 1/4.

00:06:26.206 --> 00:06:30.780
So just using this definition
of a derivative, I think we now

00:06:30.780 --> 00:06:33.340
figured out what y is equal to.

00:06:33.340 --> 00:06:41.620
y is equal to 1/4
x to the fourth.

00:06:41.620 --> 00:06:44.220
I think you might start
seeing a pattern here.

00:06:44.220 --> 00:06:46.230
Well how did we get
from x to the third to

00:06:46.230 --> 00:06:47.640
1/4 x to the fourth?

00:06:47.640 --> 00:06:51.940
Well, we increased the exponent
by 1, and whatever the new

00:06:51.940 --> 00:06:56.050
exponent is, we multiply it
times 1 over that new exponent.

00:06:56.050 --> 00:06:59.920
So let's think if we can do
a generalized rule here.

00:07:02.870 --> 00:07:05.810
Oh, and of course, plus c.

00:07:05.810 --> 00:07:08.360
I would have failed this exam.

00:07:08.360 --> 00:07:13.260
So let's make a general rule
that if I have the integral

00:07:13.260 --> 00:07:18.210
of-- well, since we already
used a, let's say-- b

00:07:18.210 --> 00:07:23.670
times x to the n dx.

00:07:23.670 --> 00:07:24.650
What is this integral?

00:07:24.650 --> 00:07:27.420
This is an integral sign.

00:07:27.420 --> 00:07:33.630
Well my new rule is, I raise
the exponent on x by 1, so it's

00:07:33.630 --> 00:07:36.880
going to be x to the n plus 1.

00:07:36.880 --> 00:07:40.980
And then I multiply x times
the inverse of this number.

00:07:40.980 --> 00:07:45.380
So times 1 over n plus 1.

00:07:45.380 --> 00:07:47.580
And of course I had that
b there all the time.

00:07:47.580 --> 00:07:50.310
And one day I'll do a more
vigorous-- more rigorous proof

00:07:50.310 --> 00:07:53.520
and maybe it will be vigorous
as well-- as to why this b

00:07:53.520 --> 00:07:56.490
just stays multiplying.

00:07:56.490 --> 00:07:59.390
Actually I don't have to do too
rigorous of a proof if you just

00:07:59.390 --> 00:08:04.030
remember how a derivative is
done, you just multiply this

00:08:04.030 --> 00:08:05.830
times the exponent minus 1.

00:08:05.830 --> 00:08:10.190
So here we multiply the
coefficient times 1 over

00:08:10.190 --> 00:08:11.530
the exponent plus 1.

00:08:11.530 --> 00:08:13.630
It's just the
inverse operation.

00:08:13.630 --> 00:08:16.460
So let's do a couple of
examples like this really fast.

00:08:16.460 --> 00:08:18.820
I have a little time left.

00:08:18.820 --> 00:08:22.420
I think the examples, at
least for me, really

00:08:22.420 --> 00:08:23.200
hit the point home.

00:08:23.200 --> 00:08:25.520
So let's say I wanted to
figure out the integral

00:08:25.520 --> 00:08:31.310
of 5 x to the seventh dx.

00:08:31.310 --> 00:08:35.850
Well, I take the exponent,
increase it by one.

00:08:35.850 --> 00:08:39.910
So I get x to the eighth, and
then I multiply the coefficient

00:08:39.910 --> 00:08:42.100
times 1 over the new exponent.

00:08:42.100 --> 00:08:45.920
So it's 5/8 x to the eighth.

00:08:45.920 --> 00:08:48.250
And if you don't trust me,
take the derivative of this.

00:08:48.250 --> 00:08:56.740
Take the derivative d/dx
of 5/8 x to the eighth.

00:08:56.740 --> 00:08:59.970
Well you multiply 8 times 5/8.

00:08:59.970 --> 00:09:04.450
Well that equals 5 x to the--
and now the new exponent will

00:09:04.450 --> 00:09:08.600
be 8 minus 1-- 5 x
to the seventh.

00:09:08.600 --> 00:09:10.880
Oh, and of course, plus c.

00:09:10.880 --> 00:09:13.090
Don't want to
forget the plus c.

00:09:13.090 --> 00:09:15.680
So I think you have a
sense of how this works.

00:09:15.680 --> 00:09:17.990
In the next presentation I'm
going to do a bunch more

00:09:17.990 --> 00:09:19.960
examples, and I'll also
show you how to kind of

00:09:19.960 --> 00:09:21.320
reverse the chain rule.

00:09:21.320 --> 00:09:23.270
And then we'll learn
integration by parts, which is

00:09:23.270 --> 00:09:25.720
essentially just reversing
the product rule.

00:09:25.720 --> 00:09:26.330
See you in the next
presentation.


Derivatives (part 9)

WEBVTT
Kind: captions
Language: en

00:00:00.970 --> 00:00:03.630
Now that you've been introduced
into some of the other

00:00:03.630 --> 00:00:05.665
functions that we can take a
derivative of, we can now

00:00:05.665 --> 00:00:07.700
apply them using the chain
and the product rule.

00:00:07.700 --> 00:00:10.770
So let's do some
fun derivatives.

00:00:10.770 --> 00:00:12.750
And I think derivatives is
all about exposure, it's

00:00:12.750 --> 00:00:13.690
all about practice.

00:00:13.690 --> 00:00:15.780
So I just encourage you to do
as much practice as possible.

00:00:15.780 --> 00:00:19.540
And it's actually in some ways
a pretty mechanical thing to do

00:00:19.540 --> 00:00:21.190
and it's easier than a lot of
the math that you've

00:00:21.190 --> 00:00:21.740
learned before.

00:00:21.740 --> 00:00:24.360
Just maybe initially
looks a little abstract.

00:00:24.360 --> 00:00:37.590
So let's say that f of x is
equal to let's say the sin

00:00:37.590 --> 00:00:43.910
of 3x to the fifth plus 2x.

00:00:43.910 --> 00:00:45.520
So what is f prime of x?

00:00:45.520 --> 00:00:48.940
What is the derivative
of this function?

00:00:48.940 --> 00:00:50.410
Well, we use the
chain rule again.

00:00:50.410 --> 00:00:52.860
We take the derivative
of the inside.

00:00:52.860 --> 00:00:54.140
So what's the derivative
of the inside?

00:00:54.140 --> 00:01:03.800
Well, that's just 5 times 3 is
15 x to the fourth plus 2.

00:01:03.800 --> 00:01:06.100
And then we take the derivative
of the larger function.

00:01:06.100 --> 00:01:08.030
In the last presentation
we learned the derivative

00:01:08.030 --> 00:01:09.700
of sin is what?

00:01:09.700 --> 00:01:10.600
It's cosine.

00:01:10.600 --> 00:01:17.370
So it's times the cosine of
this expression right here.

00:01:17.370 --> 00:01:23.010
3x to the fifth plus 2x.

00:01:23.010 --> 00:01:24.330
Pretty painless, no?

00:01:24.330 --> 00:01:25.380
Let's mix it up even more.

00:01:25.380 --> 00:01:28.030
Let's say that-- let me
switch colors just to

00:01:28.030 --> 00:01:30.700
not be monotonous.

00:01:30.700 --> 00:01:32.290
I'll pick powder blue.

00:01:32.290 --> 00:01:33.530
Very nice.

00:01:33.530 --> 00:01:36.450
Let's say that y-- and I'm
going to switch notation on

00:01:36.450 --> 00:01:38.700
purpose that you get used to
the various notations

00:01:38.700 --> 00:01:40.150
you can use.

00:01:40.150 --> 00:01:44.310
Let's say that y is equal to--
let me think of something

00:01:44.310 --> 00:01:59.580
good-- e to the x times
cosine to the fifth of x.

00:01:59.580 --> 00:02:01.460
That looks daunting to me.

00:02:01.460 --> 00:02:03.570
Let's see if we can break
it down using the product

00:02:03.570 --> 00:02:05.550
and the chain rules.

00:02:05.550 --> 00:02:07.760
We want to figure out dy/dx.

00:02:07.760 --> 00:02:10.740
We want to figure out
the rate at which y

00:02:10.740 --> 00:02:13.680
changes relative to x.

00:02:13.680 --> 00:02:15.500
Or the derivative.

00:02:15.500 --> 00:02:16.750
Find the derivative
of both sides.

00:02:16.750 --> 00:02:19.800
Well, let's use
the product rule.

00:02:19.800 --> 00:02:20.830
Well, we're going to have
to use the chain and

00:02:20.830 --> 00:02:22.130
the product rules.

00:02:22.130 --> 00:02:26.060
So first we take the derivative
of this first term, and once

00:02:26.060 --> 00:02:28.320
again we learned in the last
presentation the most amazing

00:02:28.320 --> 00:02:31.070
fact, one of the most amazing
facts in the universe that the

00:02:31.070 --> 00:02:33.670
derivative of either
the x is what?

00:02:33.670 --> 00:02:36.480
It is e to the x.

00:02:36.480 --> 00:02:38.490
Blows my mind.

00:02:38.490 --> 00:02:40.330
e to the x.

00:02:40.330 --> 00:02:42.560
Once again I've taken the
derivative, and it's

00:02:42.560 --> 00:02:43.720
the same expression.

00:02:43.720 --> 00:02:45.050
Amazing.

00:02:45.050 --> 00:02:48.070
And then I multiply it
times a second expression.

00:02:48.070 --> 00:02:51.420
Cosine to the fifth of x.

00:02:51.420 --> 00:02:54.800
And now to that I add
the derivative of the

00:02:54.800 --> 00:02:55.730
second expression.

00:02:55.730 --> 00:02:57.500
Now this will be a little
bit more interesting.

00:02:57.500 --> 00:03:03.150
So this is cosine
of x to the fifth.

00:03:03.150 --> 00:03:05.780
This is just another way
of writing cosine of

00:03:05.780 --> 00:03:08.310
x to the fifth power.

00:03:08.310 --> 00:03:09.930
And I think that'll make it a
little bit more clear, that

00:03:09.930 --> 00:03:13.050
this cosine superscript
5, this is really just

00:03:13.050 --> 00:03:14.120
cosine to the fifth x.

00:03:14.120 --> 00:03:15.980
This means cosine
of x to the fifth.

00:03:15.980 --> 00:03:17.730
So now the derivative is
a little bit clearer.

00:03:17.730 --> 00:03:19.750
We can use the chain rule--
and once again, we're just

00:03:19.750 --> 00:03:21.270
working on this right half.

00:03:21.270 --> 00:03:23.060
We take the derivative
of the inside.

00:03:23.060 --> 00:03:25.480
What's the derivative
of cosine of x?

00:03:25.480 --> 00:03:26.760
Yep you're right.

00:03:26.760 --> 00:03:28.230
Well, I don't know, I didn't
hear you so I don't know.

00:03:28.230 --> 00:03:29.175
I'll assume you're right.

00:03:29.175 --> 00:03:32.070
The derivative of cosine of x
is minus sin of x, and that's

00:03:32.070 --> 00:03:33.526
something you should memorize,
although you should prove

00:03:33.526 --> 00:03:35.085
it to yourself as well.

00:03:35.085 --> 00:03:39.360
So we take the derivative
of the inside minus sin.

00:03:39.360 --> 00:03:41.680
Derivative of cosine of x is
minus sin of x, and then we

00:03:41.680 --> 00:03:42.900
take the derivative
of the outside.

00:03:42.900 --> 00:03:44.220
We're just doing
the chain rule.

00:03:44.220 --> 00:03:50.190
So it's 5 cosine to
the fourth of x.

00:03:50.190 --> 00:03:54.640
So there we took the derivative
of this piece, and then we have

00:03:54.640 --> 00:03:57.830
to multiply times
this first piece.

00:03:57.830 --> 00:04:02.340
So that times e to the x.

00:04:02.340 --> 00:04:03.130
Interesting.

00:04:03.130 --> 00:04:05.230
You can simplify this if you
want, but you get the point.

00:04:05.230 --> 00:04:07.030
I mean simplifying it from
this point is really

00:04:07.030 --> 00:04:08.460
just kind of algebra.

00:04:08.460 --> 00:04:09.990
And I think you get the idea.

00:04:09.990 --> 00:04:12.590
Actually everything
we're doing is algebra.

00:04:12.590 --> 00:04:14.340
If you realize it looks like
something fairly complicated,

00:04:14.340 --> 00:04:16.870
but we just use the chain
and the product rules.

00:04:16.870 --> 00:04:17.850
Let's do some more.

00:04:22.030 --> 00:04:23.600
I will now switch to magenta.

00:04:26.130 --> 00:04:31.260
We want to take the derivative
dy/dx of-- let's see,

00:04:31.260 --> 00:04:32.300
some big expression.

00:04:32.300 --> 00:04:34.570
let me do something creative.

00:04:34.570 --> 00:04:49.920
Let's say the natural log
of x over 3x plus 10.

00:04:54.830 --> 00:04:57.500
So the natural log of
x over 3x plus 10.

00:04:57.500 --> 00:05:00.470
So you could use the quotient
rule if you took the time to

00:05:00.470 --> 00:05:02.770
memorize it, which I've never
taught you because it's really

00:05:02.770 --> 00:05:03.550
just the product rule.

00:05:03.550 --> 00:05:06.240
So I like to just rewrite
this as the product rule.

00:05:06.240 --> 00:05:08.310
So they're the same thing.

00:05:08.310 --> 00:05:10.280
Once again we're taking the
derivative, so I'm not going to

00:05:10.280 --> 00:05:12.830
keep rewriting this, but this
is the same thing as taking the

00:05:12.830 --> 00:05:21.825
derivative of the natural log
of x times 3x plus ten to

00:05:21.825 --> 00:05:23.700
the negative 1 power.

00:05:23.700 --> 00:05:26.695
3x plus 10 in the denominator
is the same thing as 1 over 3x

00:05:26.695 --> 00:05:29.720
plus 10, which is the same
thing as 3x plus 10 to

00:05:29.720 --> 00:05:30.990
the negative 1 power.

00:05:30.990 --> 00:05:33.610
Now we can use the combination
of the product and the

00:05:33.610 --> 00:05:35.800
chain rules, and we
can solve this sucker.

00:05:35.800 --> 00:05:38.160
So let's do it.

00:05:38.160 --> 00:05:44.150
So we take the derivative of
this first term the natural log

00:05:44.150 --> 00:05:46.142
of x-- and we learned in the
last presentation the

00:05:46.142 --> 00:05:49.330
derivative of the natural log
of x is 1/x, which is

00:05:49.330 --> 00:05:50.980
pretty cool in of itself.

00:05:50.980 --> 00:05:53.560
And we multiply that
times a second term.

00:05:53.560 --> 00:06:00.480
So time 3x plus 10 to
the negative 1 power.

00:06:00.480 --> 00:06:03.270
And to that we add the
derivative of the second term,

00:06:03.270 --> 00:06:06.430
and we're going to multiply
that times the first term.

00:06:06.430 --> 00:06:08.370
So first we're going to have
to use the chain rule.

00:06:08.370 --> 00:06:09.710
We take the derivative
of the inside.

00:06:09.710 --> 00:06:10.800
Well the derivative of
the inside's easy.

00:06:10.800 --> 00:06:12.990
The derivative of 3x x plus 10.

00:06:12.990 --> 00:06:13.900
That's just 3.

00:06:16.880 --> 00:06:18.750
And then we take the derivative
of the whole thing,

00:06:18.750 --> 00:06:22.010
so it's negative 1.

00:06:22.010 --> 00:06:25.930
That's 3 times negative
1 times that whole

00:06:25.930 --> 00:06:28.340
expression to the minus 2.

00:06:28.340 --> 00:06:30.800
3x plus 10.

00:06:30.800 --> 00:06:35.420
And of course this whole thing
times the natural log of x.

00:06:35.420 --> 00:06:36.980
We could simplify that.

00:06:36.980 --> 00:06:37.290
Let's see.

00:06:37.290 --> 00:06:40.170
This is 1/x and 3x plus
10 to negative 1.

00:06:40.170 --> 00:06:48.090
So we could rewrite this
as 1 over x 3x plus 10.

00:06:48.090 --> 00:06:49.200
Let's see.

00:06:49.200 --> 00:06:51.550
Plus 3 times minus 1.

00:06:51.550 --> 00:06:57.808
So we could say that's the
same thing as minus 3 ln of

00:06:57.808 --> 00:07:03.910
x over 3x plus 10 squared.

00:07:03.910 --> 00:07:05.620
I think you see how I
got from here to here.

00:07:05.620 --> 00:07:08.180
I just manipulated the
exponents and multiplied some

00:07:08.180 --> 00:07:11.760
numbers, et cetera, et cetera.

00:07:11.760 --> 00:07:13.080
Let's do one more.

00:07:13.080 --> 00:07:14.250
Just hit the point home.

00:07:14.250 --> 00:07:18.710
You really have the tools now
at your disposal to do a

00:07:18.710 --> 00:07:21.180
lot of derivative problems.

00:07:21.180 --> 00:07:23.640
Probably most of the derivative
problems you'll face in the

00:07:23.640 --> 00:07:26.770
first 1/2 year of calculus.

00:07:26.770 --> 00:07:28.870
I'm going to switch to green.

00:07:28.870 --> 00:07:32.720
Let's say y-- actually
I'm tired of y.

00:07:32.720 --> 00:07:44.550
Let's say that p is equal
to-- I don't know.

00:07:44.550 --> 00:07:51.510
Sin of x over cosine of x.

00:07:51.510 --> 00:07:52.996
Let's figure out what dp/dx.

00:07:55.580 --> 00:07:58.110
The rate at which
p changes to x.

00:07:58.110 --> 00:08:04.510
So once again this is the same
thing as sin of x times cosine

00:08:04.510 --> 00:08:07.600
of x to the negative 1.

00:08:07.600 --> 00:08:10.140
So we can just do the
product and chain rules.

00:08:10.140 --> 00:08:12.200
So the derivative
of the first term.

00:08:12.200 --> 00:08:16.370
Derivative of sin of x
is cosine of x, times

00:08:16.370 --> 00:08:17.990
the second term.

00:08:17.990 --> 00:08:23.270
Times cosine of x
to the minus 1.

00:08:23.270 --> 00:08:25.560
And then to that we add the
derivative of the second term.

00:08:25.560 --> 00:08:27.460
We have to use the
chain real here.

00:08:27.460 --> 00:08:29.590
So we take the derivative
of the inside.

00:08:29.590 --> 00:08:35.070
Well derivative of cosine
x is minus sin of x.

00:08:35.070 --> 00:08:37.910
And then times the
derivative of the outside.

00:08:37.910 --> 00:08:46.580
well that's minus 1 cosine of x
to the minus 2, and then we

00:08:46.580 --> 00:08:50.810
multiply that times the
first term, sin of x.

00:08:50.810 --> 00:08:51.890
So let's simplify that.

00:08:51.890 --> 00:08:56.080
So this is cosine of x
divided by cosine of x.

00:08:56.080 --> 00:08:57.320
You see how this cancels out?

00:08:57.320 --> 00:09:01.830
This is cosine of x over cosine
of x, so this is equal to 1.

00:09:01.830 --> 00:09:04.160
Cosine of x divided
cosine of x is 1.

00:09:04.160 --> 00:09:11.830
And then this minus sin cancels
out with this minus sin, and

00:09:11.830 --> 00:09:16.165
we have sin times sin
over cosine squared.

00:09:16.165 --> 00:09:17.480
So this is equal to 1.

00:09:17.480 --> 00:09:19.440
I'm going kind of fast because
I'm about to run out of time,

00:09:19.440 --> 00:09:20.720
but I think you get
what I'm doing.

00:09:20.720 --> 00:09:28.470
So this is sin squared x over
cosine squared x, which

00:09:28.470 --> 00:09:32.430
is actually equal to--
what's sin over cosine?

00:09:32.430 --> 00:09:36.800
1 plus 10 squared x.

00:09:36.800 --> 00:09:40.480
And if you know your trig
identities, that equals 1

00:09:40.480 --> 00:09:43.980
over cosine squared of x.

00:09:43.980 --> 00:09:46.890
And, of course, what we just
proved is that the derivative

00:09:46.890 --> 00:09:51.200
of the tangent of x is equal
to the secant squared of x.

00:09:51.200 --> 00:09:53.930
I hope I thoroughly confused
you in that last problem.

00:09:53.930 --> 00:09:54.730
I'll see you in the
next presentation.


Quotient rule and common derivatives

WEBVTT
Kind: captions
Language: en

00:00:00.800 --> 00:00:01.470
Welcome back.

00:00:01.470 --> 00:00:03.400
Let's do some more
derivative problems.

00:00:03.400 --> 00:00:11.910
Let's say I want to figure out
the derivative d over dx of--

00:00:11.910 --> 00:00:15.856
and let me give something that
looks a little bit different--

00:00:15.856 --> 00:00:27.070
x to the third minus 5x to the
fifth, all of that to the third

00:00:27.070 --> 00:00:41.820
power over 2x plus 5
to the fifth power.

00:00:41.820 --> 00:00:42.850
This is a parentheses.

00:00:42.850 --> 00:00:44.610
This is just saying that I want
to take the derivative of

00:00:44.610 --> 00:00:45.840
this entire expression.

00:00:45.840 --> 00:00:48.080
So you're saying Sal, we've
never learn how to do this,

00:00:48.080 --> 00:00:50.090
you have something in the
numerator, you have something

00:00:50.090 --> 00:00:51.710
in the denominator I don't
know what to do next.

00:00:51.710 --> 00:00:53.050
Well let's just rewrite this.

00:00:53.050 --> 00:00:55.310
Actually in your calculus
textbooks there's something

00:00:55.310 --> 00:00:59.350
called the quotient rule, which
I think is mildly lame, because

00:00:59.350 --> 00:01:03.290
the quotient rule is just the
product rule where you have a

00:01:03.290 --> 00:01:05.860
negative exponent and they make
it another rule, and they

00:01:05.860 --> 00:01:06.720
clutter your brain.

00:01:06.720 --> 00:01:08.470
So instead of using the
quotient rule, we're just

00:01:08.470 --> 00:01:11.780
going to rewrite this bottom
expression as a product,

00:01:11.780 --> 00:01:13.130
and then we can use
the product rule.

00:01:13.130 --> 00:01:17.500
So this is the same thing as
taking the derivative of x to

00:01:17.500 --> 00:01:22.830
the third minus 5x to the
fifth, all of that to the third

00:01:22.830 --> 00:01:28.120
power, times 2x plus 5 to what?

00:01:28.120 --> 00:01:30.380
The minus fifth power.

00:01:30.380 --> 00:01:32.060
And now we can use
the product rule.

00:01:32.060 --> 00:01:35.140
Take the derivative of the
first term-- and the derivative

00:01:35.140 --> 00:01:37.220
of the first term isn't a
joke-- you take the derivative

00:01:37.220 --> 00:01:39.190
of the inside first, let's do
the chain rule, derivative

00:01:39.190 --> 00:01:40.760
of the inside first.

00:01:40.760 --> 00:01:52.010
That is 3x squared minus 25x to
the fourth times the derivative

00:01:52.010 --> 00:01:57.700
of the outside, 3 times this
entire expression x to the

00:01:57.700 --> 00:02:02.580
third minus 5x to the fifth.

00:02:02.580 --> 00:02:04.610
And then all of that, take this
exponent down one to the

00:02:04.610 --> 00:02:08.510
squared, and then multiply
it times this whole term.

00:02:08.510 --> 00:02:14.630
So 2x plus 5 to
the minus fifth.

00:02:14.630 --> 00:02:17.240
And then to that we
add the derivative of

00:02:17.240 --> 00:02:19.570
this term, so plus.

00:02:19.570 --> 00:02:22.400
So the derivative of this term
we take the derivative of the

00:02:22.400 --> 00:02:23.600
inside, which is pretty easy.

00:02:23.600 --> 00:02:29.060
It's just 2 times the
derivative of the outside,

00:02:29.060 --> 00:02:31.065
which is minus 5.

00:02:31.065 --> 00:02:33.620
And just so you know I didn't
skip a step, the derivative of

00:02:33.620 --> 00:02:36.610
2x plus 5, the derivative of 2x
is 2, derivative of 5 is 0.

00:02:36.610 --> 00:02:38.270
So the derivative of
2x plus 5 is just 2.

00:02:38.270 --> 00:02:42.990
So it's 2 times
minus 5 2x plus 5.

00:02:42.990 --> 00:02:46.410
We just keep that the same to
the minus fifth power, and then

00:02:46.410 --> 00:02:50.360
we multiply it times this first
expression, x to the third

00:02:50.360 --> 00:02:55.340
minus 5x to the fifth
to the third power.

00:02:55.340 --> 00:02:57.270
I know that's really messy and
you'll probably not see

00:02:57.270 --> 00:02:59.570
problems this messy, but I just
wanted to show you that the

00:02:59.570 --> 00:03:02.630
product rule we learned-- it's
actually the product and the

00:03:02.630 --> 00:03:04.430
chain rule-- they can apply to
a lot of different problems,

00:03:04.430 --> 00:03:06.696
and even though you hadn't seen
something like this where you

00:03:06.696 --> 00:03:08.880
had numerator and a
denominator, you can easily

00:03:08.880 --> 00:03:12.150
rewrite what you had in
the denominator as a

00:03:12.150 --> 00:03:13.190
negative exponent.

00:03:13.190 --> 00:03:15.080
And then of course it's just
the product for when you don't

00:03:15.080 --> 00:03:19.090
have to memorize that silly
thing called the quotient rule.

00:03:19.090 --> 00:03:21.460
So with that out of the way,
I'm now going to introduce you

00:03:21.460 --> 00:03:23.450
to some common derivatives
of other functions.

00:03:23.450 --> 00:03:29.840
And these things are actually
normally included in the inside

00:03:29.840 --> 00:03:31.340
cover of your calculus book,
and they're just good to

00:03:31.340 --> 00:03:33.820
know, good things to know.

00:03:33.820 --> 00:03:36.030
And maybe in a later
presentation I'll actually

00:03:36.030 --> 00:03:37.000
prove these things.

00:03:37.000 --> 00:03:38.710
You should never take
things at face value.

00:03:38.710 --> 00:03:42.290
So you should to some degree
memorize these, although you

00:03:42.290 --> 00:03:44.640
should prove it to
yourself first.

00:03:44.640 --> 00:03:49.140
So the derivative of e to the
x-- and I find this to be

00:03:49.140 --> 00:03:52.855
amazing. e shows up all sorts
of crazy places in mathematics,

00:03:52.855 --> 00:03:56.770
and it's you know the strange
number 2.7 whatever, whatever

00:03:56.770 --> 00:03:58.560
and it has all sorts of
strange properties.

00:03:58.560 --> 00:04:02.660
And I think this is one of the
most bizarre properties of e.

00:04:02.660 --> 00:04:03.930
The derivative of e to the x.

00:04:03.930 --> 00:04:07.880
So if I want to figure out
the slope of any point along

00:04:07.880 --> 00:04:11.820
the curve e to the x-- this
just might blow your mind.

00:04:11.820 --> 00:04:13.610
I think the more you think
about it, the more it'll blow

00:04:13.610 --> 00:04:17.010
your mind-- is e to the x.

00:04:17.010 --> 00:04:17.950
That's amazing.

00:04:17.950 --> 00:04:21.810
At any point along the curve
e to the x, the slope of

00:04:21.810 --> 00:04:25.900
that point is e to the x.

00:04:25.900 --> 00:04:27.230
Just to hit the point home.

00:04:27.230 --> 00:04:29.150
I'm diverging, a little bit.

00:04:29.150 --> 00:04:34.640
But if I said f of x is
equal to e to the x, right?

00:04:34.640 --> 00:04:39.590
And let's say f of 2 is
equal to e squared.

00:04:39.590 --> 00:04:44.540
And I asked you, friend-- I
don't know your name-- what

00:04:44.540 --> 00:04:52.010
is the slope of e to the x
at the point 2,e squared.

00:04:52.010 --> 00:04:54.540
And you could say Sal,
the slope at that

00:04:54.540 --> 00:04:57.510
point is e squared.

00:04:57.510 --> 00:05:01.720
That blows my mind that it's a
function where the slope at

00:05:01.720 --> 00:05:04.960
any point on that line is
equal to the function.

00:05:04.960 --> 00:05:07.570
And it's e. e shows up
all sorts of places.

00:05:07.570 --> 00:05:11.070
I might do a whole series of
presentations called the

00:05:11.070 --> 00:05:15.530
magic of e, because e shows
up all over the place.

00:05:15.530 --> 00:05:17.360
Well I don't want to diverge
too much, so that's

00:05:17.360 --> 00:05:18.530
pretty amazing.

00:05:18.530 --> 00:05:21.760
Next I'm going to show you what
I think is probably the second

00:05:21.760 --> 00:05:24.980
most amazing derivative-- and I
don't think this has been fully

00:05:24.980 --> 00:05:27.740
explored in mathematics yet,
because this also blows my

00:05:27.740 --> 00:05:34.870
mind-- is that the derivative
of the natural log of x, right.

00:05:34.870 --> 00:05:37.980
So the natural log is just the
logarithm with base e, and I

00:05:37.980 --> 00:05:39.440
hope you remember
your logarithms.

00:05:39.440 --> 00:05:41.220
So what's the derivative
of the natural log of x?

00:05:41.220 --> 00:05:43.480
So once again this
is e related.

00:05:43.480 --> 00:05:46.430
Well it's 1/x.

00:05:46.430 --> 00:05:48.730
That also blows my mind.

00:05:48.730 --> 00:05:50.800
Because think about it.

00:05:50.800 --> 00:05:52.270
Let's draw a bunch
of functions.

00:05:52.270 --> 00:06:00.940
If I said the derivative
of x to the minus 3 is

00:06:00.940 --> 00:06:04.460
minus 3x to the minus 4.

00:06:04.460 --> 00:06:08.690
The derivative of x to
the minus 2 is minus

00:06:08.690 --> 00:06:14.480
2x to the minus 3.

00:06:14.480 --> 00:06:20.850
The derivative of x to
the minus 1 is minus

00:06:20.850 --> 00:06:23.870
1 x to the minus 2.

00:06:23.870 --> 00:06:30.750
The derivative of x to the 0--
well this is just 1, right?

00:06:30.750 --> 00:06:35.920
The derivative of x to the 0 is
just 1, so the derivative is 0.

00:06:35.920 --> 00:06:39.382
The derivative of x is 1,
derivative of x squared

00:06:39.382 --> 00:06:42.470
is 2x and so on, right?

00:06:42.470 --> 00:06:44.420
So it's interesting.

00:06:44.420 --> 00:06:48.130
We have this pattern from all
the derivatives of all of the

00:06:48.130 --> 00:06:50.940
of kind of the exponents in
increasing order where you go

00:06:50.940 --> 00:06:56.360
from x to the minus 4 x to the
minus 3, x to the minus 2 and

00:06:56.360 --> 00:06:59.100
then there's no x to
the minus 1 here.

00:06:59.100 --> 00:07:01.920
We go straight to x to the 0.

00:07:01.920 --> 00:07:06.040
What happened to x the minus 1?

00:07:06.040 --> 00:07:07.590
What happened to this?

00:07:07.590 --> 00:07:10.290
What function's derivative
is x to the minus 1?

00:07:10.290 --> 00:07:11.860
This is bizarre to me.

00:07:11.860 --> 00:07:12.490
Where did it go?

00:07:12.490 --> 00:07:16.260
And it turns out that
it's a natural log.

00:07:16.260 --> 00:07:19.260
This I still think about before
I go to bed sometimes because

00:07:19.260 --> 00:07:21.830
it is kind of mind blowing.

00:07:21.830 --> 00:07:23.370
And later in another
presentation I might

00:07:23.370 --> 00:07:24.240
actually prove this to you.

00:07:24.240 --> 00:07:26.370
But just to know that this is
true, that the derivative of

00:07:26.370 --> 00:07:29.810
the natural log of x is 1/x
I think is mind blowing.

00:07:29.810 --> 00:07:31.870
And so for now you can
just memorize it.

00:07:31.870 --> 00:07:33.150
But both of these
are mind blowing.

00:07:33.150 --> 00:07:35.830
The derivative of e to the x is
e to the x, and the derivative

00:07:35.830 --> 00:07:38.780
of the natural log of x is 1/x.

00:07:38.780 --> 00:07:41.700
And I'll just do a couple of
more just to present them to

00:07:41.700 --> 00:07:43.950
you, and then in the next
presentation we'll actually use

00:07:43.950 --> 00:07:46.680
them using the product rule and
the chain rule and et

00:07:46.680 --> 00:07:47.170
cetera, et cetera.

00:07:47.170 --> 00:07:50.120
And you might want to rewatch
this and memorize them.

00:07:50.120 --> 00:07:53.760
I want to clear image.

00:07:53.760 --> 00:07:54.910
OK.

00:07:54.910 --> 00:07:57.920
And now I'll just do the basic
trig functions, and you should

00:07:57.920 --> 00:07:59.240
memorize these as well.

00:07:59.240 --> 00:08:02.510
The derivative of sin of x--
this is pretty easy to

00:08:02.510 --> 00:08:05.140
remember-- is cosine of x.

00:08:05.140 --> 00:08:06.850
So the slope at any point
along the [? line ?]

00:08:06.850 --> 00:08:09.260
sin of x is actually the
cosine of that point.

00:08:09.260 --> 00:08:10.050
That's also interesting.

00:08:10.050 --> 00:08:11.460
One day I'm going to do this
holographically because I

00:08:11.460 --> 00:08:14.060
think that might not be
sinking in properly.

00:08:14.060 --> 00:08:19.610
The derivative of cosine
of x is minus sin of x.

00:08:19.610 --> 00:08:21.710
There are good to memorize
though, because you'll be

00:08:21.710 --> 00:08:24.500
able to recall is quickly
on a test and then use it.

00:08:24.500 --> 00:08:31.620
And then finally the derivative
of tan of x is equal to 1 over

00:08:31.620 --> 00:08:35.900
cosine square of x which you
could also write as the

00:08:35.900 --> 00:08:38.580
secant squared of x.

00:08:38.580 --> 00:08:41.170
You might want to memorize
these now, and actually I

00:08:41.170 --> 00:08:43.790
encourage you to explore these
things, I encourage you to

00:08:43.790 --> 00:08:45.980
graph each of these functions.

00:08:45.980 --> 00:08:49.160
Graph a function, graph its
derivative and look at them,

00:08:49.160 --> 00:08:53.080
and really intuitively
understand why the derivative

00:08:53.080 --> 00:08:57.040
function actually does
describe the slope of

00:08:57.040 --> 00:08:57.820
the original function.

00:08:57.820 --> 00:08:59.610
And actually I'll probably
do a presentation on that.

00:08:59.610 --> 00:09:01.065
But I'm almost out of time
in this presentation,

00:09:01.065 --> 00:09:03.950
so just memorize these.

00:09:03.950 --> 00:09:06.380
And memorize the derivative of
e to the x, e to the x, and

00:09:06.380 --> 00:09:08.940
the natural log of x is 1/x.

00:09:08.940 --> 00:09:11.720
And in the next presentation
we're going to start mixing and

00:09:11.720 --> 00:09:13.910
matching all of these
functions, and we can use the

00:09:13.910 --> 00:09:17.490
product and chain rule on them
to solve kind of arbitrarily

00:09:17.490 --> 00:09:19.520
complex derivatives.

00:09:19.520 --> 00:09:23.320
Between what we've just seen,
we could probably solve 95% of

00:09:23.320 --> 00:09:26.100
the derivative problems
you'll see on say the

00:09:26.100 --> 00:09:28.140
calculus AP test.

00:09:28.140 --> 00:09:28.810
I'll see you in the
next presentation.


Product rule

WEBVTT
Kind: captions
Language: en

00:00:00.990 --> 00:00:02.920
Welcome back.

00:00:02.920 --> 00:00:06.530
I'm now going to introduce
you to a new tool for

00:00:06.530 --> 00:00:08.370
solving derivatives.

00:00:08.370 --> 00:00:12.010
Really between this rule, which
is the product rule, and the

00:00:12.010 --> 00:00:16.000
chain rule and just knowing a
lot of function derivatives,

00:00:16.000 --> 00:00:19.330
you'll be ready to tackle
almost any derivative problem.

00:00:19.330 --> 00:00:20.420
Let's start with
the chain rule.

00:00:20.420 --> 00:00:31.810
Let's say that f of x is equal
to h of x times g of x.

00:00:31.810 --> 00:00:32.780
This is the product rule.

00:00:32.780 --> 00:00:36.080
In the chain rule it was f of
x is equal to h of g of x.

00:00:36.080 --> 00:00:36.310
Right?

00:00:36.310 --> 00:00:37.730
I don't know if you
remember that.

00:00:37.730 --> 00:00:40.980
In this case, f of x is equal
to h of x times g of x.

00:00:40.980 --> 00:00:45.570
If that's the case, then f
prime of x is equal to the

00:00:45.570 --> 00:00:48.940
derivative of the first
function times a second

00:00:48.940 --> 00:00:55.450
function plus the first
function times the derivative

00:00:55.450 --> 00:00:57.030
of the second function.

00:00:57.030 --> 00:00:57.990
Pretty straightforward.

00:00:57.990 --> 00:00:59.350
Let's apply it.

00:00:59.350 --> 00:01:02.660
Let's say that-- I don't like
this brown color, let me pick

00:01:02.660 --> 00:01:05.190
something more pleasant.

00:01:05.190 --> 00:01:08.250
Maybe mauve.

00:01:08.250 --> 00:01:17.270
Let's say that f of x is equal
to 5x to the fifth minus x to

00:01:17.270 --> 00:01:29.850
the seventh times 20x squared
plus 3x to them mine 7.

00:01:29.850 --> 00:01:31.610
So one way we could have
done it, we could just

00:01:31.610 --> 00:01:32.520
multiply this out.

00:01:32.520 --> 00:01:36.310
This wouldn't be too bad, and
then take the derivative

00:01:36.310 --> 00:01:37.200
like any polynomial.

00:01:37.200 --> 00:01:40.400
But let's use this product rule
that I've just shown you.

00:01:40.400 --> 00:01:43.470
So the product rules that says,
let me take the derivative of

00:01:43.470 --> 00:01:46.890
the first expression, or h of x
if we wanted to map

00:01:46.890 --> 00:01:48.240
it into this rule.

00:01:48.240 --> 00:01:50.360
The derivative of that is
pretty straightforward.

00:01:50.360 --> 00:01:52.470
5 times 5 is 25.

00:01:52.470 --> 00:01:57.160
25x to the fourth, right?

00:01:57.160 --> 00:02:01.480
Then minus 7, x to the sixth.

00:02:01.480 --> 00:02:04.780
We're just going to multiply it
times this second expression,

00:02:04.780 --> 00:02:07.290
doing nothing different to it.

00:02:07.290 --> 00:02:09.980
Maybe I'll just do it
in a different color.

00:02:09.980 --> 00:02:18.920
Times 20x plus 3x minus 7.

00:02:18.920 --> 00:02:24.270
And then to that we will
add the derivative of

00:02:24.270 --> 00:02:27.110
this second function.

00:02:27.110 --> 00:02:35.000
The derivative of that second
function is 40x minus

00:02:35.000 --> 00:02:38.590
21x to the minus eighth.

00:02:38.590 --> 00:02:41.570
And that times this
first function.

00:02:41.570 --> 00:02:42.870
I guess I'll switch
back to mauve, I think

00:02:42.870 --> 00:02:44.140
you get the point.

00:02:44.140 --> 00:02:51.480
5x to the fifth minus
x to the seventh.

00:02:51.480 --> 00:02:54.660
All we did here was we said OK,
f of x is made of these two

00:02:54.660 --> 00:02:56.730
expressions and they are
multiplied by each other.

00:02:56.730 --> 00:02:58.930
If I want to take the
derivative of it, I take the

00:02:58.930 --> 00:03:03.250
derivative of the first one and
multiply it by the second one.

00:03:03.250 --> 00:03:05.530
And then I add that to the
derivative of the second

00:03:05.530 --> 00:03:07.360
one and multiply it
by the first one.

00:03:07.360 --> 00:03:10.050
Let's do some more examples
and I think that will

00:03:10.050 --> 00:03:11.860
hit the point home.

00:03:11.860 --> 00:03:14.752
Clear image.

00:03:14.752 --> 00:03:18.850
Change the colors and
I'm back in business.

00:03:18.850 --> 00:03:21.440
Let me think of a good problem.

00:03:21.440 --> 00:03:23.410
Let me do another one like
this, and then I'll actually

00:03:23.410 --> 00:03:27.790
introduce ones and the product
rule and the chain rule.

00:03:27.790 --> 00:03:40.630
So let's say that f of x is
equal to 10x to the third plus

00:03:40.630 --> 00:03:53.180
5x squared minus 7 times
20x to the eighth minus 7.

00:03:53.180 --> 00:03:56.970
Then we say f prime of x,
what's the derivative of

00:03:56.970 --> 00:03:58.710
this first expression.

00:03:58.710 --> 00:04:05.900
It's 30x squared plus 10x.

00:04:05.900 --> 00:04:09.290
And I just multiply it times
this expression, right?

00:04:09.290 --> 00:04:13.570
20x to the eighth minus 7.

00:04:13.570 --> 00:04:16.240
And I add that to the
derivative of this second

00:04:16.240 --> 00:04:21.130
expression, this is all on one
line but I ran out of space,

00:04:21.130 --> 00:04:24.600
160x to the seventh, right?

00:04:24.600 --> 00:04:27.490
8 times 20 is 160.

00:04:27.490 --> 00:04:29.520
And then the derivative
of 7 is zero.

00:04:29.520 --> 00:04:33.440
So it's just 160x to
the seventh times this

00:04:33.440 --> 00:04:35.060
first expression.

00:04:35.060 --> 00:04:42.720
10x to the third plus 5x
squared minus seven.

00:04:42.720 --> 00:04:43.290
There we go.

00:04:43.290 --> 00:04:44.300
And you could simplify it.

00:04:44.300 --> 00:04:46.190
You could multiply this out
if you wanted or you could

00:04:46.190 --> 00:04:48.590
distribute this out if you
wanted, maybe try to

00:04:48.590 --> 00:04:49.730
condense the terms.

00:04:49.730 --> 00:04:51.420
But that's really just algebra.

00:04:51.420 --> 00:04:54.030
So this is using
the product rule.

00:04:54.030 --> 00:04:55.880
I'm going to do one more
example where I'll show you,

00:04:55.880 --> 00:04:58.140
I'm going to use the product
and the chain rule and

00:04:58.140 --> 00:05:02.170
I think this will
optimally confuse you.

00:05:02.170 --> 00:05:03.225
I want to make sure
I have some space.

00:05:07.910 --> 00:05:09.260
Here I'm going to use a
slightly different notation.

00:05:09.260 --> 00:05:12.110
Instead of saying f of x and
then what's f prime of x, I'm

00:05:12.110 --> 00:05:27.110
going to say y is equal to x
squared plus 2x to the fifth

00:05:27.110 --> 00:05:40.200
times 3x to the minus three
plus x squared to the minus 7.

00:05:40.200 --> 00:05:44.480
And I want to find the rate at
which y changes relative to x.

00:05:44.480 --> 00:05:48.130
So I want to find dy over dx.

00:05:48.130 --> 00:05:49.950
This is just like, if this
was f of x, it's just

00:05:49.950 --> 00:05:52.360
like saying f prime of x.

00:05:52.360 --> 00:05:53.290
This is just a [UNINTELLIGIBLE]

00:05:53.290 --> 00:05:54.470
notation.

00:05:54.470 --> 00:05:55.560
So what do I do in
the chain rule?

00:05:55.560 --> 00:05:58.630
First I want the
derivative of this term.

00:05:58.630 --> 00:06:02.460
Let me use colors to make
it not too confusing.

00:06:02.460 --> 00:06:05.950
So what's the derivative
of this term?

00:06:05.950 --> 00:06:08.540
We are going to use
the chain rule first.

00:06:08.540 --> 00:06:15.990
So we take the derivative of
the inside which is 2x plus 2

00:06:15.990 --> 00:06:18.790
and multiply times
the derivative of the

00:06:18.790 --> 00:06:20.330
larger expression.

00:06:20.330 --> 00:06:26.920
But we keep x squared plus 3x
there so it's times 5 times

00:06:26.920 --> 00:06:28.470
something to the fourth.

00:06:28.470 --> 00:06:33.320
And that something is
x squared plus 2x.

00:06:33.320 --> 00:06:36.460
So there we took the derivative
of this first term right here

00:06:36.460 --> 00:06:38.300
and then the product rules says
we take the derivative of the

00:06:38.300 --> 00:06:40.740
first term, we just multiply
it by the second term.

00:06:40.740 --> 00:06:49.120
So the second term is just 3x
to the minus 3 plus x squared

00:06:49.120 --> 00:06:51.080
and all that to the minus 7.

00:06:51.080 --> 00:06:57.240
We did that and then to that we
add plus the derivative of this

00:06:57.240 --> 00:06:59.870
second term times
this first term.

00:06:59.870 --> 00:07:01.280
We're going to use the
chain rule again.

00:07:01.280 --> 00:07:02.750
What's the derivative
of the second term?

00:07:02.750 --> 00:07:04.890
I'll switch back to
the light blue.

00:07:04.890 --> 00:07:07.840
Light blue means the derivative
of one of the terms.

00:07:07.840 --> 00:07:11.320
So we take the derivative of
the inside, the derivative of

00:07:11.320 --> 00:07:18.710
inside is minus 3 times 3 is
minus 9, x go down one to

00:07:18.710 --> 00:07:23.210
the minus 4, plus 2x.

00:07:23.210 --> 00:07:26.180
And now we take the derivative
of the whole thing.

00:07:26.180 --> 00:07:34.090
Times minus 7 times something
to the minus 8, and that

00:07:34.090 --> 00:07:36.360
something is this inside.

00:07:36.360 --> 00:07:40.280
3x to the minus 3
plus x squared.

00:07:40.280 --> 00:07:42.770
And then we multiply this
thing, this whole thing which

00:07:42.770 --> 00:07:44.870
is the derivative of the second
term times the first term.

00:07:47.730 --> 00:07:53.010
Times, and I'm just going to
keep going, times x squared

00:07:53.010 --> 00:07:57.190
plus 2x to the fifth.

00:07:57.190 --> 00:07:59.710
So this is a really, I
mean you might want to

00:07:59.710 --> 00:08:00.730
simplify at this point.

00:08:00.730 --> 00:08:02.480
You can take this minus
7 and multiply it

00:08:02.480 --> 00:08:03.490
out and all of that.

00:08:03.490 --> 00:08:05.350
But I think this
gives you the idea.

00:08:05.350 --> 00:08:08.850
And if you have to multiply
this out and then do the

00:08:08.850 --> 00:08:10.220
derivative if it's just a
polynomial, this would

00:08:10.220 --> 00:08:11.080
take you forever.

00:08:11.080 --> 00:08:14.110
But using the chain rule,
you're actually able to, even

00:08:14.110 --> 00:08:16.250
though we ended up with a
pretty complicated answer,

00:08:16.250 --> 00:08:17.110
we got the right answer.

00:08:17.110 --> 00:08:20.230
And now we could actually
evaluate the slope of this very

00:08:20.230 --> 00:08:22.990
complicated function at any
point just by substituting the

00:08:22.990 --> 00:08:25.190
point into this fairly
complicated expression.

00:08:25.190 --> 00:08:27.830
But at least we could do it.

00:08:27.830 --> 00:08:29.970
I think you're going to find
that the chain and the product

00:08:29.970 --> 00:08:33.230
rules become even more useful
once we start doing derivatives

00:08:33.230 --> 00:08:35.970
of expressions other
than polynomials.

00:08:35.970 --> 00:08:38.620
I'm going to teach you about
trigonometric functions and

00:08:38.620 --> 00:08:42.710
natural log and logarithm
and exponential functions.

00:08:42.710 --> 00:08:44.460
Actually, I'll probably do that
in the next presentation.

00:08:44.460 --> 00:08:47.240
So I will see you soon.


Even More Chain Rule

WEBVTT
Kind: captions
Language: en

00:00:01.260 --> 00:00:03.796
Now that you've seen some
examples of the chain rule

00:00:03.796 --> 00:00:06.930
in use, I think the actual
definition of the chain rule

00:00:06.930 --> 00:00:08.770
might be more digestible now.

00:00:08.770 --> 00:00:11.550
So let me give you the actual
definition of the chain rule.

00:00:11.550 --> 00:00:27.340
Let's say I have a function f
of x and it equals h of g of x.

00:00:27.340 --> 00:00:29.640
And you remember all this
from composite functions.

00:00:29.640 --> 00:00:33.350
So the chain rule just says
that the derivative of f of x

00:00:33.350 --> 00:00:39.500
or f prime of x is equal to
the derivative of this inner

00:00:39.500 --> 00:00:48.370
function, g prime of x times
the derivative of this h

00:00:48.370 --> 00:00:52.600
function, h prime of x.

00:00:52.600 --> 00:00:54.105
But it's not going to
be just h prime of x.

00:00:54.105 --> 00:00:59.070
It's going to be h
prime of g of x.

00:00:59.070 --> 00:01:01.750
So let's apply that to some
examples like we were

00:01:01.750 --> 00:01:04.290
doing before, and I think
it'll make some sense.

00:01:04.290 --> 00:01:17.870
So let's say we had f of x is
equal to x squared plus 5x

00:01:17.870 --> 00:01:24.080
plus 3, all of this
to the fifth power.

00:01:24.080 --> 00:01:27.580
So in this example, what's
h of x, what's g of x, and

00:01:27.580 --> 00:01:28.440
you know what f of x is.

00:01:28.440 --> 00:01:33.240
Well let's say g of x would
be this inner function.

00:01:33.240 --> 00:01:40.010
So we would say-- let me pick a
different color-- g of x here

00:01:40.010 --> 00:01:47.260
is x squared plus 5x plus 3.

00:01:47.260 --> 00:01:49.480
It's the stuff, f g of x.

00:01:52.260 --> 00:01:55.715
Well h of g of x is this whole
thing, so what would h of x be?

00:02:01.750 --> 00:02:04.840
This is h of g of x, but
h of x would just be x

00:02:04.840 --> 00:02:06.940
to the fifth, right?

00:02:06.940 --> 00:02:10.530
Because this expression as you
took this entire g of x and you

00:02:10.530 --> 00:02:12.370
put it in for x right here.

00:02:12.370 --> 00:02:15.630
I think that make sense if you
take this entire expression and

00:02:15.630 --> 00:02:17.720
you substitute x here for this
entire expression you

00:02:17.720 --> 00:02:19.810
get this expression.

00:02:19.810 --> 00:02:30.280
And this shows that this
is equal to h of g of x.

00:02:30.280 --> 00:02:33.450
If you just take this blue part
and substitute it for x, you

00:02:33.450 --> 00:02:35.410
get this entire expression.

00:02:35.410 --> 00:02:38.860
So the chain rule just tells us
that the derivative of this,

00:02:38.860 --> 00:02:40.980
that f prime of x-- and I have
a feeling I'm going to run out

00:02:40.980 --> 00:02:45.570
of space-- f prime of x-- well
actually before I do anything,

00:02:45.570 --> 00:02:46.950
let's figure out the
derivatives of g

00:02:46.950 --> 00:02:48.470
of x and h of x.

00:02:48.470 --> 00:02:52.700
g of x of g prime of x-- let me
draw a little line here to

00:02:52.700 --> 00:02:55.530
divide it out, I know I'm
running out of space.

00:02:55.530 --> 00:03:03.530
So g prime of x is
equal to 2x plus 5.

00:03:06.200 --> 00:03:09.290
2x plus 5, and then derivative
3 is just 0, right?

00:03:09.290 --> 00:03:17.150
And the derivative of h of
x? h prime of x is equal

00:03:17.150 --> 00:03:22.410
to 5 x to the fourth.

00:03:22.410 --> 00:03:25.270
So the chain rule just says
that the derivative of this

00:03:25.270 --> 00:03:30.530
entire composite function
is just-- let me just

00:03:30.530 --> 00:03:31.240
write it down here.

00:03:31.240 --> 00:03:34.850
I'm doing this to
optimally confuse you.

00:03:34.850 --> 00:03:40.050
The derivative of this entire
function is just g prime of x.

00:03:40.050 --> 00:03:42.450
Well we figured out with g
prime of x is here, it's 2x

00:03:42.450 --> 00:03:55.260
plus 5 times h times
h prime of g of x.

00:03:55.260 --> 00:03:56.310
So what's h prime of g of x?

00:03:56.310 --> 00:03:59.750
Well h prime of x is 5x to
the fourth, but we want

00:03:59.750 --> 00:04:01.800
h prime of g of x.

00:04:01.800 --> 00:04:11.330
So h prime of g of x
would equal 5 times g

00:04:11.330 --> 00:04:14.190
of x to the fourth.

00:04:14.190 --> 00:04:17.310
And we know what g of x is,
it's this whole thing.

00:04:17.310 --> 00:04:21.970
So it would be times 5, and
this whole thing x squared

00:04:21.970 --> 00:04:26.990
plus 5x plus 3, all that
to the fourth power.

00:04:26.990 --> 00:04:29.815
I think I have truly, truly
confused you, so I'm going

00:04:29.815 --> 00:04:33.630
to try to do a couple
of more examples.

00:04:33.630 --> 00:04:36.310
Clear this.

00:04:36.310 --> 00:04:37.560
OK.

00:04:37.560 --> 00:04:39.320
Let me write it up here again.

00:04:39.320 --> 00:04:55.870
So if we say that f of x is
equal to h of g of x, then f

00:04:55.870 --> 00:05:07.055
prime of x is equal to g prime
of x times h prime of g of x.

00:05:09.790 --> 00:05:10.955
So I'll do another example.

00:05:15.270 --> 00:05:28.750
Let's say that g of x is equal
to x to the seventh minus

00:05:28.750 --> 00:05:33.150
3x to the ninth is 3.

00:05:33.150 --> 00:05:44.870
And let's say that h of x is
equal to-- let's do something

00:05:44.870 --> 00:05:46.260
reasonably straightforward.

00:05:46.260 --> 00:05:55.570
Let's say h of x is
x to the minus 10.

00:05:55.570 --> 00:06:00.430
So what is f of x? f of x is
just h of g of x, and this

00:06:00.430 --> 00:06:03.660
should be a bit of a reminder
from composite functions.

00:06:03.660 --> 00:06:04.310
So let's see.

00:06:04.310 --> 00:06:15.995
h of g of x would just be-- you
take g of x and you substitute

00:06:15.995 --> 00:06:25.090
it for x here, so it would just
be this expression, x to the

00:06:25.090 --> 00:06:31.670
seventh minus 3x to the minus
third, and then all of that

00:06:31.670 --> 00:06:33.810
to the minus 10th power.

00:06:33.810 --> 00:06:35.720
So this is our f of x.

00:06:35.720 --> 00:06:38.700
And this is of course equal to
f of x, right, because f of

00:06:38.700 --> 00:06:41.350
x is equal to h of g of x.

00:06:41.350 --> 00:06:44.050
I know this very confusing,
but bear with me.

00:06:44.050 --> 00:06:45.650
Maybe you have to watch
the video twice and it'll

00:06:45.650 --> 00:06:47.130
start making more sense.

00:06:47.130 --> 00:06:49.770
Well we want to now figure
out what f prime of x is.

00:06:54.760 --> 00:06:58.570
Well the chain rule tells us
all it is, is we take the

00:06:58.570 --> 00:07:00.930
derivative of g of x, right?

00:07:00.930 --> 00:07:03.240
So the derivative
of g of x is what?

00:07:03.240 --> 00:07:04.450
That's easy.

00:07:04.450 --> 00:07:06.190
Or hopefully it's easy by now.

00:07:06.190 --> 00:07:12.280
Derivative of g of x is 7x to
the sixth, and minus 3 times

00:07:12.280 --> 00:07:16.720
minus 3 is plus 9x
to the minus 4.

00:07:16.720 --> 00:07:20.790
I just took minus 3 and went
down 1, so that's g prime of x.

00:07:23.350 --> 00:07:31.130
And then times h
prime of g of x.

00:07:31.130 --> 00:07:34.550
Well what's h prime of x?

00:07:34.550 --> 00:07:35.010
That's easy.

00:07:35.010 --> 00:07:39.890
That's just minus 10
times x to the minus 11.

00:07:39.890 --> 00:07:43.050
But we want to do h
prime of g of x.

00:07:43.050 --> 00:07:45.590
So instead of having an x here,
we're going to substitute that

00:07:45.590 --> 00:07:48.320
x with the entire g
of x expression.

00:07:48.320 --> 00:07:55.760
So this is just times 10 time
something to the minus eleven,

00:07:55.760 --> 00:07:57.600
and that something
is just g of x.

00:08:00.310 --> 00:08:05.780
x to the seventh, minus 3x
access to the minus 3.

00:08:05.780 --> 00:08:10.780
And there's our answer. f prime
of x is the derivative of kind

00:08:10.780 --> 00:08:15.300
of the inner function, g of x,
times the derivative of the

00:08:15.300 --> 00:08:19.420
outer function, but instead of
it just being applied to x it'd

00:08:19.420 --> 00:08:23.960
be applied to the entire g of x
instead of an x being here.

00:08:23.960 --> 00:08:25.470
Maybe I've confused you more.

00:08:25.470 --> 00:08:27.680
Let me do one quick example
just to show you that you

00:08:27.680 --> 00:08:30.940
don't have to kind of do this
whole h of g of every time.

00:08:33.790 --> 00:08:48.980
So if I have f of x is equal to
5 times minus x to the eighth,

00:08:48.980 --> 00:08:54.180
plus x to the minus eighth,
all of that over to

00:08:54.180 --> 00:08:57.630
the fifth power.

00:08:57.630 --> 00:09:00.610
If I want to figure out f prime
of x I just take the derivative

00:09:00.610 --> 00:09:05.480
of this inner function I guess
I could call it, so that's

00:09:05.480 --> 00:09:12.480
minus 8x to the seventh minus
8x-- because it's just take the

00:09:12.480 --> 00:09:18.580
negative 8-- to the minus
ninth, times the derivative

00:09:18.580 --> 00:09:19.920
of this larger function.

00:09:19.920 --> 00:09:27.740
So 5 times 5 is 25 times
something to the fourth.

00:09:27.740 --> 00:09:30.440
And that something is just
going to be this expression

00:09:30.440 --> 00:09:35.730
minus x to the eighth plus
x to the minus eighth.

00:09:35.730 --> 00:09:36.290
And we're done.

00:09:36.290 --> 00:09:37.100
You could simplify it.

00:09:37.100 --> 00:09:40.080
You could multiply this 25 out
and do et cetera, et cetera.

00:09:40.080 --> 00:09:42.540
Hopefully this gives you more
of an intuition of what the

00:09:42.540 --> 00:09:45.730
chain rule is all about, and
I'm going to do a lot more

00:09:45.730 --> 00:09:47.910
examples in the next couple
of presentations as well.

00:09:47.910 --> 00:09:48.250
See you soon.


Chain Rule Examples

WEBVTT
Kind: captions
Language: en

00:00:00.960 --> 00:00:02.790
I'm now going to do a
bunch more examples

00:00:02.790 --> 00:00:04.120
using the chain rule.

00:00:04.120 --> 00:00:05.320
So let's see.

00:00:05.320 --> 00:00:05.920
Once again.

00:00:05.920 --> 00:00:13.930
If I had f of x is equal to,
let's see, I don't like this

00:00:13.930 --> 00:00:16.100
tool that I'm using now,
let's have one of these.

00:00:16.100 --> 00:00:28.390
f of x is equal to, say, x to
the third plus 2x squared

00:00:28.390 --> 00:00:32.600
minus, let's say, minus
x to the negative 2.

00:00:32.600 --> 00:00:34.960
We haven't put any negative
exponents in yet, but I

00:00:34.960 --> 00:00:38.060
think you'll see that
the same patterns apply.

00:00:38.060 --> 00:00:43.520
And all of that to, let's
say, the minus seven.

00:00:43.520 --> 00:00:47.077
We want to figure out what
f prime of x, what the

00:00:47.077 --> 00:00:48.260
derivative of f of x is.

00:00:48.260 --> 00:00:50.540
So this might seem very
complicated and daunting to

00:00:50.540 --> 00:00:53.220
you, and obviously to take
this entire polynomial to

00:00:53.220 --> 00:00:55.470
the negative seventh power
would take you forever.

00:00:55.470 --> 00:00:57.590
But using the chain rule, we
can do it quite quickly.

00:00:57.590 --> 00:01:00.610
So the first thing we want to
do, is we want to take the

00:01:00.610 --> 00:01:03.460
derivative of the inner
function, I guess

00:01:03.460 --> 00:01:04.420
you could call it.

00:01:04.420 --> 00:01:06.630
We want to take the
derivative of this.

00:01:06.630 --> 00:01:08.990
And what's the derivative of x
to the third plus 2x squared

00:01:08.990 --> 00:01:10.770
minus x to the negative 2?

00:01:10.770 --> 00:01:11.600
Well, we know how to do that.

00:01:11.600 --> 00:01:13.790
That was the first type
of derivatives we

00:01:13.790 --> 00:01:14.210
learned how to do.

00:01:14.210 --> 00:01:22.210
It's 3x squared and 2 times 2,
plus 4x to the first, or just

00:01:22.210 --> 00:01:27.270
4x, and then here, with a
negative exponent, we do

00:01:27.270 --> 00:01:28.120
the exact same thing.

00:01:28.120 --> 00:01:31.320
We say negative 2 times
negative 1, right, there's a 1

00:01:31.320 --> 00:01:33.450
here, we don't write it down.

00:01:33.450 --> 00:01:40.360
So negative 2 times negative 1
is plus 2 x to the, and then we

00:01:40.360 --> 00:01:43.210
decrease the exponent by
1, so it's x to the

00:01:43.210 --> 00:01:45.530
negative 3, right?

00:01:45.530 --> 00:01:47.980
So we figured out what the
derivative of the inside is,

00:01:47.980 --> 00:01:53.990
and then we just multiply
that, that whole thing, times

00:01:53.990 --> 00:01:56.750
the derivative of kind of
the entire expression.

00:01:56.750 --> 00:02:01.210
So then that'll be, we
take the minus 7, let me

00:02:01.210 --> 00:02:02.270
do a different color.

00:02:02.270 --> 00:02:05.800
So this is the entire thing.

00:02:05.800 --> 00:02:13.620
So then we take minus 7, so
it's times minus 7, this

00:02:13.620 --> 00:02:16.130
whole expression, I'm
going to run out of space.

00:02:16.130 --> 00:02:22.370
x to the third plus 2x squared
minus x to the minus 2.

00:02:22.370 --> 00:02:24.400
That's minus x to the minus 2.

00:02:24.400 --> 00:02:26.900
And all of that, we just
decrease this exponent

00:02:26.900 --> 00:02:29.980
by 1, to the minus 8.

00:02:29.980 --> 00:02:33.040
So let me write it all down
a little bit neater now.

00:02:33.040 --> 00:02:38.230
So we get f prime of x as the
derivative of f of x is equal

00:02:38.230 --> 00:02:52.500
to 3x squared plus 4x plus 2x
to the minus third power, I

00:02:52.500 --> 00:02:53.310
don't know why did that.

00:02:53.310 --> 00:02:55.190
That's minus 3.

00:02:55.190 --> 00:03:04.450
Times minus seven times x to
the third plus 2x squared minus

00:03:04.450 --> 00:03:09.710
x to the minus two, all of that
to the negative eight power.

00:03:09.710 --> 00:03:10.970
And we could simplify
it little bit.

00:03:10.970 --> 00:03:14.640
Maybe we could just multiply
this minus 7, times, we could

00:03:14.640 --> 00:03:16.005
distribute it across
this expression.

00:03:16.005 --> 00:03:23.510
So we'd say, that equals minus
7, so this equals minus 21 x

00:03:23.510 --> 00:03:34.260
squared, minus 28 x minus
14x to the negative 3.

00:03:34.260 --> 00:03:43.980
All of that times x to the
third plus 2 x squared minus x

00:03:43.980 --> 00:03:48.950
to the minus 2 to the minus 8.

00:03:48.950 --> 00:03:49.950
So there we did it.

00:03:49.950 --> 00:03:53.990
We took this, what I would say
is a very complicated function,

00:03:53.990 --> 00:03:57.290
and using the chain rule and
just the basic rules we had

00:03:57.290 --> 00:03:59.260
introduced a couple of
presentations ago, we were able

00:03:59.260 --> 00:04:01.030
to find the derivative of it.

00:04:01.030 --> 00:04:03.650
And now, if we wanted to, for
whatever application, we could

00:04:03.650 --> 00:04:07.320
find the slope of this function
at any point x by just

00:04:07.320 --> 00:04:13.910
substituting that point into
this equation, and we'll get

00:04:13.910 --> 00:04:15.120
the slope at that point.

00:04:15.120 --> 00:04:17.400
Let me do a slightly harder
one, to show you that the chain

00:04:17.400 --> 00:04:20.400
rule, you can kind of go
arbitrarily deep in

00:04:20.400 --> 00:04:20.969
the chain rule.

00:04:30.010 --> 00:04:30.610
OK.

00:04:30.610 --> 00:04:34.010
So let's say I had, let
me see if I can write it

00:04:34.010 --> 00:04:35.000
a little bit thinner.

00:04:35.000 --> 00:04:39.340
If I had f of x, I don't know
if you can see that, I'm going

00:04:39.340 --> 00:04:40.560
to do it a little fatter.

00:04:40.560 --> 00:04:46.480
f of x is equal to, I want to
make it a little bit more

00:04:46.480 --> 00:04:48.520
complicated this time.

00:04:48.520 --> 00:05:03.420
3x to the minus 2 plus 5 x to
the third minus 7x, all of that

00:05:03.420 --> 00:05:07.620
to the fifth, and then this
whole expression to

00:05:07.620 --> 00:05:08.900
the third power.

00:05:08.900 --> 00:05:11.560
So I imagine you saying, Sal,
you're starting to go nuts,

00:05:11.560 --> 00:05:13.010
this is going to
take us forever.

00:05:13.010 --> 00:05:14.630
Well, I'll show you, using
the chain rule, it will

00:05:14.630 --> 00:05:16.500
not take that long.

00:05:16.500 --> 00:05:20.200
So the way I think about
it, so-- f prime of x,

00:05:20.200 --> 00:05:22.480
f prime of x equals.

00:05:22.480 --> 00:05:25.360
I start off kind with
the innermost function.

00:05:25.360 --> 00:05:28.210
So let me see if I can
use colors to make it

00:05:28.210 --> 00:05:28.820
a little bit simpler.

00:05:28.820 --> 00:05:34.430
Let's take the derivative of
this innermost function first.

00:05:34.430 --> 00:05:35.940
Actually, let me give
you the big picture.

00:05:35.940 --> 00:05:40.820
We want to find the derivative
of the innermost function, and

00:05:40.820 --> 00:05:42.860
then a little bit bigger,
and then a little bit

00:05:42.860 --> 00:05:43.890
more big than that.

00:05:43.890 --> 00:05:46.980
I know that's not precise
mathematical terms, but

00:05:46.980 --> 00:05:49.310
you'll get the point when
I show you this example.

00:05:49.310 --> 00:05:51.860
So first we'll do this
inner function, this

00:05:51.860 --> 00:05:52.860
inner expression.

00:05:52.860 --> 00:05:54.520
And the derivative of
that's pretty easy, right?

00:05:54.520 --> 00:06:00.270
It's 15x squared
minus 7, right?

00:06:00.270 --> 00:06:02.020
that was pretty
straightforward.

00:06:02.020 --> 00:06:06.150
And now we're going to want
to multiply that times this

00:06:06.150 --> 00:06:08.010
entire derivative here.

00:06:08.010 --> 00:06:10.280
So let me circle that in
a different-- so then

00:06:10.280 --> 00:06:11.750
we want to do this.

00:06:11.750 --> 00:06:15.240
We're going to multiply that
times this entire derivative.

00:06:15.240 --> 00:06:20.140
Well, that's just times 5.

00:06:20.140 --> 00:06:24.420
And we just pretend like this
is just an x here, right?

00:06:24.420 --> 00:06:26.795
Because the derivative
of x to the fifth is 5x

00:06:26.795 --> 00:06:28.150
to the fourth, right?

00:06:28.150 --> 00:06:30.590
But instead of an x, we have
this whole expression, 5x

00:06:30.590 --> 00:06:31.680
to the third minus 7x.

00:06:31.680 --> 00:06:33.130
So we'll write that.

00:06:33.130 --> 00:06:37.360
5x to the third minus 7x.

00:06:37.360 --> 00:06:41.520
Now the exponent here
goes down by one.

00:06:41.520 --> 00:06:44.540
So it's 5 times 5x to the
third minus 7x, all that

00:06:44.540 --> 00:06:46.070
to the fourth power.

00:06:46.070 --> 00:06:48.330
So we figured out the
derivative of this so far, and

00:06:48.330 --> 00:06:51.646
then we want to figure out the
derivative of this, so we'll

00:06:51.646 --> 00:06:53.000
add it, right, because we're
trying to figure the derivative

00:06:53.000 --> 00:06:54.880
of this entire expression.

00:06:54.880 --> 00:06:56.120
So this is an easy one.

00:06:56.120 --> 00:07:00.020
Let me draw that in
a different color.

00:07:00.020 --> 00:07:02.870
So we want the
derivative of this.

00:07:02.870 --> 00:07:06.450
So that's negative 2 times
3, so that's negative

00:07:06.450 --> 00:07:10.430
6x to the minus three.

00:07:10.430 --> 00:07:11.610
So what have we done so far?

00:07:11.610 --> 00:07:16.270
We've so far figured out the
derivative of this entire

00:07:16.270 --> 00:07:18.540
expression, right?

00:07:18.540 --> 00:07:20.850
The derivative of that
entire expression using

00:07:20.850 --> 00:07:25.170
the chain rule is this.

00:07:25.170 --> 00:07:26.910
And now, we're almost done.

00:07:26.910 --> 00:07:28.290
We just have to multiply that.

00:07:28.290 --> 00:07:30.630
So I'm going to just, I've run
out of space on that line, but

00:07:30.630 --> 00:07:32.170
let's just assume that
the line continues.

00:07:32.170 --> 00:07:33.770
So that's times.

00:07:33.770 --> 00:07:35.360
And now we just take the
derivative of kind of

00:07:35.360 --> 00:07:37.410
this whole big thing.

00:07:37.410 --> 00:07:40.346
And now it's going to be the
derivative of, I'm going

00:07:40.346 --> 00:07:42.580
to use this brown color.

00:07:42.580 --> 00:07:45.950
So it's a whole big expression
to the third power, right?

00:07:45.950 --> 00:07:50.520
So that becomes times 3 times
the whole expression, right?

00:07:50.520 --> 00:07:55.270
That's 3 times, now I'm going
to write the whole thing, 3x to

00:07:55.270 --> 00:08:06.670
the minus 2 plus 5 x the third
minus 7x, that to fifth, and

00:08:06.670 --> 00:08:10.220
then you decrement this by
1, to the second power.

00:08:10.220 --> 00:08:12.600
That was an ultraconfusing
example, and this is probably

00:08:12.600 --> 00:08:15.850
the hardest chain rule problem
you'll see in a lot of

00:08:15.850 --> 00:08:17.340
the questions you'll
have on your test.

00:08:17.340 --> 00:08:18.450
You see, it wasn't
that difficult.

00:08:18.450 --> 00:08:21.200
We just kind of went to the
smallest possible function, and

00:08:21.200 --> 00:08:22.760
actually the smallest possible
function would have been one of

00:08:22.760 --> 00:08:25.710
these terms, but we just found
the derivative of this, which

00:08:25.710 --> 00:08:30.690
was 15 x squared minus 7, and
then we just used the principle

00:08:30.690 --> 00:08:33.420
that the derivative of kind of
a function is just the

00:08:33.420 --> 00:08:36.960
derivative of each of its
parts-- well, actually, the

00:08:36.960 --> 00:08:40.400
derivative of-- we figured out
the derivative of this inner

00:08:40.400 --> 00:08:43.464
piece, which was 15x squared
minus 7, and then we multiplied

00:08:43.464 --> 00:08:47.350
it times the derivative of this
slightly larger piece, which is

00:08:47.350 --> 00:08:52.210
5 times this entire expression
to the fourth, then we added it

00:08:52.210 --> 00:08:54.980
to the derivative of
3x to the minus 2.

00:08:54.980 --> 00:08:57.470
And then that whole thing, and
actually I should put a big

00:08:57.470 --> 00:09:00.810
parentheses around here, that
whole thing, we multiply it

00:09:00.810 --> 00:09:04.180
times the derivative of
this larger expression.

00:09:04.180 --> 00:09:07.630
I think I might have confused
you, so I apologize if I have,

00:09:07.630 --> 00:09:09.810
and in the next presentation
I'm going to just do a bunch

00:09:09.810 --> 00:09:13.000
more chain rule problems, and
at some point, it should

00:09:13.000 --> 00:09:14.670
start to make sense to you.

00:09:14.670 --> 00:09:18.210
I think it's just a matter
of seeing example, after

00:09:18.210 --> 00:09:19.940
example, after example.

00:09:19.940 --> 00:09:22.380
I'll see you into the next
presentation, and I apologize

00:09:22.380 --> 00:09:24.440
if I have confused you.


The Chain Rule

WEBVTT
Kind: captions
Language: en

00:00:00.960 --> 00:00:01.930
Welcome back.

00:00:01.930 --> 00:00:05.460
I'm now going to do some more
examples of a bit of a review

00:00:05.460 --> 00:00:07.300
of some of the derivatives
that we've been seeing.

00:00:07.300 --> 00:00:09.450
And then I'll introduce you to
something called the chain rule

00:00:09.450 --> 00:00:13.080
which expands the universe of
the types of functions we can

00:00:13.080 --> 00:00:14.290
take the derivatives of.

00:00:14.290 --> 00:00:18.540
So in the last presentation, I
showed you how to function if I

00:00:18.540 --> 00:00:30.270
had f of x is equal to 10x to
the seventh plus 6x to the

00:00:30.270 --> 00:00:39.330
third plus 15x minus
x to the 16th.

00:00:39.330 --> 00:00:42.160
To take the derivative of this
entire function, we take just

00:00:42.160 --> 00:00:43.840
the derivatives of each
of the pieces, right?

00:00:43.840 --> 00:00:45.640
Because you can add them up.

00:00:45.640 --> 00:00:51.560
So f prime of x in this
example, is equal to-- and

00:00:51.560 --> 00:00:52.755
I think you get the hang
of it at this point.

00:00:52.755 --> 00:00:54.150
It's actually fairly
straightforward.

00:00:54.150 --> 00:00:56.480
We take the 7, multiply
it by the 10.

00:00:56.480 --> 00:01:01.620
So we get 70x, then
1 degree less.

00:01:01.620 --> 00:01:10.940
So 70x to the sixth plus
18x squared plus 15.

00:01:10.940 --> 00:01:13.010
We can kind of view this
as x to the 1, right?

00:01:13.010 --> 00:01:16.920
So it's 1 times 15
times x to the 0.

00:01:16.920 --> 00:01:17.550
Which is 1.

00:01:17.550 --> 00:01:23.840
So that's just 15 minus
16x to the 15th.

00:01:23.840 --> 00:01:25.330
And I don't want you to
lose sight of what we're

00:01:25.330 --> 00:01:26.090
actually doing here.

00:01:26.090 --> 00:01:27.870
What is f prime of x?

00:01:27.870 --> 00:01:32.770
This is the function the tells
us the slope of any point

00:01:32.770 --> 00:01:35.450
x, along the curve f of x.

00:01:35.450 --> 00:01:37.860
It's a pretty
interesting thing.

00:01:37.860 --> 00:01:40.795
Let me just draw to maybe give
you a little bit of intuition.

00:01:45.300 --> 00:01:47.490
I don't know what the slope
of f of x really looks like.

00:01:47.490 --> 00:01:49.560
And actually, let's pretend
like this isn't f of x.

00:01:49.560 --> 00:01:51.360
Let's pretend like this
is just some arbitrary

00:01:51.360 --> 00:01:53.270
function I'm drawing.

00:01:53.270 --> 00:01:56.210
If this is f of x, just some
curve that does all sorts of

00:01:56.210 --> 00:02:01.430
crazy things, f prime of x
tells me the slope at any

00:02:01.430 --> 00:02:02.810
point along that line.

00:02:02.810 --> 00:02:09.950
So if I wanted to know the
slope at this point right here,

00:02:09.950 --> 00:02:13.060
I could use the derivative
function to figure out the

00:02:13.060 --> 00:02:15.120
slope of the tangent line.

00:02:15.120 --> 00:02:18.740
The tangent line is something
like that right there.

00:02:18.740 --> 00:02:21.930
Or if I wanted to figure out
the slope at this point,

00:02:21.930 --> 00:02:23.890
once again I'd use the
derivative function.

00:02:23.890 --> 00:02:26.340
And it would tell me the
slope of the tangent

00:02:26.340 --> 00:02:27.100
line at that point.

00:02:27.100 --> 00:02:29.150
Which would be
something like that.

00:02:29.150 --> 00:02:30.960
So it's a pretty useful thing.

00:02:30.960 --> 00:02:34.900
And once I give you all the
tools to analytically solve a

00:02:34.900 --> 00:02:36.560
whole host of derivatives, then
we'll actually do a bunch

00:02:36.560 --> 00:02:39.140
of word problems and
applications of derivatives.

00:02:39.140 --> 00:02:41.270
And I think you'll see that
it's a really, really,

00:02:41.270 --> 00:02:42.700
really useful concept.

00:02:42.700 --> 00:02:43.600
So let's move on.

00:02:43.600 --> 00:02:45.960
I think you get the idea of
how to do these derivatives

00:02:45.960 --> 00:02:48.610
of polynomials.

00:02:48.610 --> 00:02:49.760
Let me erase this.

00:02:49.760 --> 00:02:52.650
I'm actually using a
different tool now.

00:02:52.650 --> 00:02:56.110
So I think it might
be a bit easier.

00:02:56.110 --> 00:02:57.630
Let's see, someone
was calling me.

00:02:57.630 --> 00:03:00.520
But you're more important so
I will not answer the phone.

00:03:08.440 --> 00:03:11.640
I'm going to introduce you--
this tool doesn't have, I

00:03:11.640 --> 00:03:14.330
don't think it has a
straight up eraser.

00:03:14.330 --> 00:03:15.230
Actually, maybe let's see.

00:03:15.230 --> 00:03:16.350
If I do it like this.

00:03:27.480 --> 00:03:28.400
Oh let me see.

00:03:30.900 --> 00:03:32.150
No that doesn't work.

00:03:32.150 --> 00:03:36.080
Let me just erase like this,
the old-fashioned way.

00:03:39.530 --> 00:03:41.000
You just have to bear with me.

00:03:41.000 --> 00:03:45.280
And then once I finish erasing,
I will show you the chain rule.

00:03:45.280 --> 00:03:45.900
This is good.

00:03:45.900 --> 00:03:48.410
It feels like I'm a real
teacher with a real chalkboard

00:03:48.410 --> 00:03:49.490
and a real eraser now.

00:03:53.530 --> 00:03:56.750
This is a lot cleaner than a
normal chalkboard as well.

00:03:56.750 --> 00:04:00.960
Bear with me, almost there.

00:04:00.960 --> 00:04:02.630
I'll figure out a faster
way to do this over the

00:04:02.630 --> 00:04:04.590
next couple of videos.

00:04:04.590 --> 00:04:05.210
It's pretty sad.

00:04:05.210 --> 00:04:07.670
I'm showing you how to do
derivatives in calculus, but

00:04:07.670 --> 00:04:10.910
I don't know how to erase
a faster way than this.

00:04:14.270 --> 00:04:15.560
There, we're done.

00:04:15.560 --> 00:04:15.900
OK.

00:04:15.900 --> 00:04:18.290
So now I'm going show you how
to solve the derivatives of a

00:04:18.290 --> 00:04:21.360
slightly more complicated
type of a function.

00:04:21.360 --> 00:04:23.080
It's actually not
more complicated.

00:04:23.080 --> 00:04:24.450
It's just different.

00:04:24.450 --> 00:04:42.890
So let's say f of x is equal to
2x plus 3 to the fifth power.

00:04:42.890 --> 00:04:45.220
And I want to figure out
the derivative of this.

00:04:45.220 --> 00:04:46.320
We're going to use something
called the chain rule.

00:04:46.320 --> 00:04:49.020
Because one thing we could do,
we could just multiply out 2x

00:04:49.020 --> 00:04:50.600
plus 3 to the fifth power.

00:04:50.600 --> 00:04:54.420
And if you've ever done
that, you know it's a pain.

00:04:54.420 --> 00:04:55.540
So that's not something
we'd want to do.

00:04:55.540 --> 00:04:57.280
So we're going to use something
called the chain rule.

00:04:57.280 --> 00:04:59.390
And I'm just going to give you
a bunch of examples before I

00:04:59.390 --> 00:05:01.060
even show you the definition
of the chain rule.

00:05:01.060 --> 00:05:02.330
Because I think this is
something that you just

00:05:02.330 --> 00:05:04.260
have to learn by example.

00:05:04.260 --> 00:05:07.110
So the chain rule just tells us
that the derivative of let's

00:05:07.110 --> 00:05:10.820
say this function right here.

00:05:10.820 --> 00:05:14.100
You take the derivative of the
subfunctions, and then you can

00:05:14.100 --> 00:05:16.000
take a derivative of
the entire function.

00:05:16.000 --> 00:05:17.290
I'll tell you that formally.

00:05:17.290 --> 00:05:19.010
But I think when you introduce
it formally, it gets

00:05:19.010 --> 00:05:19.890
more confusing.

00:05:19.890 --> 00:05:23.455
So what I do, I just take the
derivative of 2x plus 3 first.

00:05:23.455 --> 00:05:25.000
And actually let me use colors.

00:05:25.000 --> 00:05:26.730
I think that might simplify it.

00:05:26.730 --> 00:05:29.510
So I take the derivative
of 2x plus 3.

00:05:29.510 --> 00:05:31.350
What's the derivative
of 2x plus 3?

00:05:31.350 --> 00:05:32.130
Well you know that.

00:05:32.130 --> 00:05:36.470
It's just the derivative
of 2x, which is 2.

00:05:36.470 --> 00:05:37.860
And then the
derivative of 3 is 0.

00:05:37.860 --> 00:05:41.300
So the derivative of
2x plus 3 is just 2.

00:05:41.300 --> 00:05:45.030
And then I'm going to multiply
that times the derivative

00:05:45.030 --> 00:05:46.150
of the whole thing.

00:05:46.150 --> 00:05:48.930
And I just pretend like
2x plus 3 is just like

00:05:48.930 --> 00:05:51.060
a variable by itself.

00:05:51.060 --> 00:05:54.620
So then what's the derivative
of x to the fifth?

00:05:54.620 --> 00:05:56.630
Well the derivative of x to the
fifth-- I'm going to do that in

00:05:56.630 --> 00:06:00.366
a different color-- the
derivative of x to the

00:06:00.366 --> 00:06:05.590
fifth is 5x to the fourth.

00:06:05.590 --> 00:06:13.270
So it'll be 5 times
something to the fourth.

00:06:13.270 --> 00:06:15.760
But here we didn't take the
derivative of x the fifth.

00:06:15.760 --> 00:06:18.010
We took the derivative of
2x plus 3 to the fifth.

00:06:18.010 --> 00:06:20.280
So we just put the 2x
plus 3 there instead.

00:06:24.230 --> 00:06:26.190
So what did we do here?

00:06:26.190 --> 00:06:29.290
We went in the inside of
the function, and we took

00:06:29.290 --> 00:06:29.860
the derivative here.

00:06:29.860 --> 00:06:32.570
And the derivative of
2x plus 3 was just 2.

00:06:32.570 --> 00:06:35.570
And then we multiplied
it by the derivative of

00:06:35.570 --> 00:06:36.460
the greater function.

00:06:36.460 --> 00:06:38.980
And we just pretended like the
2x plus 3 was a variable.

00:06:38.980 --> 00:06:40.120
It was like x.

00:06:40.120 --> 00:06:44.410
So instead of 5x to the
fourth, we got 5 times

00:06:44.410 --> 00:06:46.170
2x plus 3 to the fourth.

00:06:46.170 --> 00:06:51.540
And if we just simplify that,
f prime of x is equal to

00:06:51.540 --> 00:06:58.220
2 times 5 is 10; 10 times
2x plus 3 to the fourth.

00:06:58.220 --> 00:07:01.650
That was a lot simpler than
multiplying out 2x plus 3 to

00:07:01.650 --> 00:07:04.260
the fifth power, and then doing
the derivatives the old way.

00:07:04.260 --> 00:07:06.080
I know this was probably a
little confusing to you,

00:07:06.080 --> 00:07:09.270
so I'm going to try to do
a couple more examples.

00:07:09.270 --> 00:07:25.570
Let's say I had g of x is equal
to x-squared plus 2x plus

00:07:25.570 --> 00:07:30.620
3 to the eighth power.

00:07:30.620 --> 00:07:36.430
So g prime of x is going to
equal-- well what did we say?

00:07:36.430 --> 00:07:38.090
We take the derivative
of the inside.

00:07:38.090 --> 00:07:40.260
This is called the chain rule.

00:07:40.260 --> 00:07:41.410
What's the derivative
of the inside?

00:07:41.410 --> 00:07:48.250
It's 2x plus 2 plus 0, right?

00:07:48.250 --> 00:07:50.190
And then we take the derivative
of the whole thing.

00:07:50.190 --> 00:07:53.100
And we just pretend like this
whole expression, x-squared

00:07:53.100 --> 00:07:57.540
plus 2x plus 3 is just kind
of like the variable x.

00:07:57.540 --> 00:07:59.590
We know that the derivative
of x to the eighth is

00:07:59.590 --> 00:08:01.520
8x to the seventh.

00:08:01.520 --> 00:08:07.410
So it'll be 8 times
something to the seventh.

00:08:07.410 --> 00:08:10.580
And that something is this
entire expression here, 8 times

00:08:10.580 --> 00:08:15.660
x-squared plus 2x plus 3.

00:08:15.660 --> 00:08:17.040
I hope I didn't
confuse you too much.

00:08:17.040 --> 00:08:18.790
And you can simplify
this more in any way.

00:08:18.790 --> 00:08:21.900
Because it's 2x plus 2 times
8 times x-squared plus 2x

00:08:21.900 --> 00:08:23.035
plus 3 to the seventh.

00:08:23.035 --> 00:08:26.300
To multiply this out, or
to multiply this out is

00:08:26.300 --> 00:08:27.800
a huge pain as you know.

00:08:27.800 --> 00:08:30.840
But we could simplify
a little bit.

00:08:30.840 --> 00:08:33.630
Let me draw a divider here.

00:08:33.630 --> 00:08:44.526
We could say that that equals 8
times 2x, 16x plus 16 times--

00:08:44.526 --> 00:08:48.300
I'm making it really messy--
x-squared plus 2x plus 3

00:08:48.300 --> 00:08:50.610
to the seventh power.

00:08:50.610 --> 00:08:51.870
I hope I didn't
confuse you too much.

00:08:51.870 --> 00:08:53.920
In the next presentation, I'm
just going to do a ton of

00:08:53.920 --> 00:08:55.500
examples using the chain rule.

00:08:55.500 --> 00:08:57.850
And I think the more examples
you see, it's going to

00:08:57.850 --> 00:08:58.470
hit the point home.

00:08:58.470 --> 00:09:00.440
And then after I've done a
bunch of examples, then I'm

00:09:00.440 --> 00:09:01.960
going to give you a
formal definition.

00:09:01.960 --> 00:09:04.190
I think that's actually an
easier way to digest the chain

00:09:04.190 --> 00:09:06.970
rule than giving you the formal
definition first, and then

00:09:06.970 --> 00:09:08.440
showing you a bunch
of examples.

00:09:08.440 --> 00:09:10.520
So I'll see you in the
next presentation.