Critical points introduction

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:03.446
I've drawn a crazy looking
function here in yellow.

00:00:03.446 --> 00:00:04.820
And what I want
to think about is

00:00:04.820 --> 00:00:07.370
when this function takes
on the maximum values

00:00:07.370 --> 00:00:08.674
and minimum values.

00:00:08.674 --> 00:00:10.090
And for the sake
of this video, we

00:00:10.090 --> 00:00:12.130
can assume that the
graph of this function

00:00:12.130 --> 00:00:14.410
just keeps getting lower
and lower and lower

00:00:14.410 --> 00:00:16.900
as x becomes more and more
negative, and lower and lower

00:00:16.900 --> 00:00:19.430
and lower as x goes
beyond the interval

00:00:19.430 --> 00:00:21.430
that I've depicted
right over here.

00:00:21.430 --> 00:00:24.580
So what is the maximum value
that this function takes on?

00:00:24.580 --> 00:00:25.780
Well we can eyeball that.

00:00:25.780 --> 00:00:30.030
It looks like it's at that
point right over there.

00:00:30.030 --> 00:00:36.390
So we would call this
a global maximum.

00:00:36.390 --> 00:00:36.890
the?

00:00:36.890 --> 00:00:40.020
Function never takes on
a value larger than this.

00:00:40.020 --> 00:00:46.450
So we could say that we have a
global maximum at the point x0.

00:00:46.450 --> 00:00:52.900
Because f of of x0 is
greater than, or equal to,

00:00:52.900 --> 00:00:56.350
f of x, for any other
x in the domain.

00:00:56.350 --> 00:00:59.140
And that's pretty obvious,
when you look at it like this.

00:00:59.140 --> 00:01:01.070
Now do we have a
global minimum point,

00:01:01.070 --> 00:01:02.650
the way that I've drawn it?

00:01:02.650 --> 00:01:03.310
Well, no.

00:01:03.310 --> 00:01:08.070
This function can take an
arbitrarily negative values.

00:01:08.070 --> 00:01:09.672
It approaches
negative infinity as x

00:01:09.672 --> 00:01:10.880
approaches negative infinity.

00:01:10.880 --> 00:01:12.420
It approaches
negative infinity as x

00:01:12.420 --> 00:01:14.080
approaches positive infinity.

00:01:14.080 --> 00:01:16.100
So we have-- let me
write this down--

00:01:16.100 --> 00:01:24.190
we have no global minimum.

00:01:24.190 --> 00:01:25.690
Now let me ask you a question.

00:01:25.690 --> 00:01:29.630
Do we have local
minima or local maxima?

00:01:29.630 --> 00:01:32.370
When I say minima, it's
just the plural of minimum.

00:01:32.370 --> 00:01:35.830
And maxima is just
the plural of maximum.

00:01:35.830 --> 00:01:40.430
So do we have a local minima
here, or local minimum here?

00:01:40.430 --> 00:01:43.300
Well, a local minimum,
you could imagine means

00:01:43.300 --> 00:01:45.180
that that value of the
function at that point

00:01:45.180 --> 00:01:47.284
is lower than the
points around it.

00:01:47.284 --> 00:01:49.700
So right over here, it looks
like we have a local minimum.

00:01:54.310 --> 00:01:56.800
And I'm not giving a very
rigorous definition here.

00:01:56.800 --> 00:01:58.330
But one way to
think about it is,

00:01:58.330 --> 00:02:02.940
we can say that we have a
local minimum point at x1,

00:02:02.940 --> 00:02:06.690
as if we have a region
around x1, where f of x1

00:02:06.690 --> 00:02:10.910
is less than an f of x for any x
in this region right over here.

00:02:10.910 --> 00:02:13.060
And it's pretty easy
to eyeball, too.

00:02:13.060 --> 00:02:16.950
This is a low point for any
of the values of f around it,

00:02:16.950 --> 00:02:18.540
right over there.

00:02:18.540 --> 00:02:21.000
Now do we have any
other local minima?

00:02:21.000 --> 00:02:22.570
Well it doesn't look like we do.

00:02:22.570 --> 00:02:25.360
Now what about local maxima?

00:02:25.360 --> 00:02:31.275
Well this one right over
here-- let me do it in purple,

00:02:31.275 --> 00:02:33.150
I don't want to get
people confused, actually

00:02:33.150 --> 00:02:35.670
let me do it in this color--
this point right over here

00:02:35.670 --> 00:02:38.590
looks like a local maximum.

00:02:38.590 --> 00:02:40.620
Not lox, that would have
to deal with salmon.

00:02:40.620 --> 00:02:46.960
Local maximum, right over there.

00:02:46.960 --> 00:02:53.790
So we could say at the point
x1, or sorry, at the point x2,

00:02:53.790 --> 00:02:56.890
we have a local
maximum point at x2.

00:02:56.890 --> 00:03:01.210
Because f of x2 is larger
than f of x for any

00:03:01.210 --> 00:03:03.814
x around a
neighborhood around x2.

00:03:03.814 --> 00:03:04.980
I'm not being very rigorous.

00:03:04.980 --> 00:03:07.280
But you can see it
just by looking at it.

00:03:07.280 --> 00:03:08.260
So that's fair enough.

00:03:08.260 --> 00:03:12.260
We've identified all of the
maxima and minima, often called

00:03:12.260 --> 00:03:15.630
the extrema, for this function.

00:03:15.630 --> 00:03:18.270
Now how can we identify
those, if we knew something

00:03:18.270 --> 00:03:20.434
about the derivative
of the function?

00:03:20.434 --> 00:03:21.850
Well, let's look
at the derivative

00:03:21.850 --> 00:03:25.390
at each of these points.

00:03:25.390 --> 00:03:27.370
So at this first
point, right over here,

00:03:27.370 --> 00:03:30.220
if I were to try to
visualize the tangent line--

00:03:30.220 --> 00:03:32.570
let me do that in a
better color than brown.

00:03:32.570 --> 00:03:35.240
If I were to try to
visualize the tangent line,

00:03:35.240 --> 00:03:37.130
it would look
something like that.

00:03:37.130 --> 00:03:38.890
So the slope here is 0.

00:03:38.890 --> 00:03:43.180
So we would say that f
prime of x0 is equal to 0.

00:03:43.180 --> 00:03:45.800
The slope of the tangent
line at this point is 0.

00:03:45.800 --> 00:03:47.140
What about over here?

00:03:47.140 --> 00:03:49.970
Well, once again,
the tangent line

00:03:49.970 --> 00:03:52.400
would look something like that.

00:03:52.400 --> 00:03:59.390
So once again, we would say
f prime at x1 is equal to 0.

00:03:59.390 --> 00:04:01.030
What about over here?

00:04:01.030 --> 00:04:04.910
Well, here the tangent line
is actually not well defined.

00:04:04.910 --> 00:04:06.705
We have a positive
slope going into it,

00:04:06.705 --> 00:04:09.180
and then it immediately jumps
to being a negative slope.

00:04:09.180 --> 00:04:14.034
So over here, f prime
of x2 is not defined.

00:04:14.034 --> 00:04:15.200
Let me just write undefined.

00:04:18.517 --> 00:04:21.100
So we have an interesting-- and
once again, I'm not rigorously

00:04:21.100 --> 00:04:24.100
proving it to you, I just want
you to get the intuition here.

00:04:24.100 --> 00:04:27.870
We see that if we have
some type of an extrema--

00:04:27.870 --> 00:04:29.820
and we're not
talking about when x

00:04:29.820 --> 00:04:31.497
is at an endpoint
of an interval,

00:04:31.497 --> 00:04:33.830
just to be clear what I'm
talking about when I'm talking

00:04:33.830 --> 00:04:36.032
about x as an endpoint
of an interval.

00:04:36.032 --> 00:04:37.740
We're saying, let's
say that the function

00:04:37.740 --> 00:04:42.905
is where you have an
interval from there.

00:04:42.905 --> 00:04:45.280
So let's say a function starts
right over there, and then

00:04:45.280 --> 00:04:45.871
keeps going.

00:04:45.871 --> 00:04:48.370
This would be a maximum point,
but it would be an end point.

00:04:48.370 --> 00:04:50.203
We're not talking about
endpoints right now.

00:04:50.203 --> 00:04:52.960
We're talking about when
we have points in between,

00:04:52.960 --> 00:04:56.000
or when our interval
is infinite.

00:04:56.000 --> 00:04:59.540
So we're not talking
about points like that,

00:04:59.540 --> 00:05:00.866
or points like this.

00:05:00.866 --> 00:05:02.615
We're talking about
the points in between.

00:05:05.690 --> 00:05:09.040
So if you have a point
inside of an interval,

00:05:09.040 --> 00:05:10.720
it's going to be a
minimum or maximum.

00:05:10.720 --> 00:05:12.430
And we see the intuition here.

00:05:12.430 --> 00:05:26.030
If you have-- so non-endpoint
min or max at, let's say,

00:05:26.030 --> 00:05:27.930
x is equal to a.

00:05:27.930 --> 00:05:30.660
So if you know that you have
a minimum or a maximum point,

00:05:30.660 --> 00:05:32.650
at some point x is
equal to a, and x

00:05:32.650 --> 00:05:36.399
isn't the endpoint
of some interval,

00:05:36.399 --> 00:05:37.940
this tells you
something interesting.

00:05:37.940 --> 00:05:39.356
Or at least we
have the intuition.

00:05:39.356 --> 00:05:42.710
We see that the derivative
at x is equal to a

00:05:42.710 --> 00:05:44.640
is going to be equal to 0.

00:05:44.640 --> 00:05:49.450
Or the derivative at x is equal
to a is going to be undefined.

00:05:52.020 --> 00:05:54.010
And we see that in
each of these cases.

00:05:54.010 --> 00:05:58.300
Derivative is 0, derivative
is 0, derivative is undefined.

00:05:58.300 --> 00:06:03.600
And we have a word for these
points where the derivative is

00:06:03.600 --> 00:06:06.600
either 0, or the
derivative is undefined.

00:06:06.600 --> 00:06:08.113
We called them critical points.

00:06:13.640 --> 00:06:16.130
So for the sake
of this function,

00:06:16.130 --> 00:06:25.380
the critical points are,
we could include x sub 0,

00:06:25.380 --> 00:06:27.580
we could include x sub 1.

00:06:27.580 --> 00:06:30.330
At x sub 0 and x sub
1, the derivative is 0.

00:06:30.330 --> 00:06:34.670
And x sub 2, where the
function is undefined.

00:06:34.670 --> 00:06:38.390
Now, so if we have a
non-endpoint minimum or maximum

00:06:38.390 --> 00:06:41.935
point, then it's going
to be a critical point.

00:06:41.935 --> 00:06:43.560
But can we say it
the other way around?

00:06:43.560 --> 00:06:47.120
If we find a critical point,
where the derivative is 0,

00:06:47.120 --> 00:06:48.830
or the derivative is
undefined, is that

00:06:48.830 --> 00:06:52.670
going to be a maximum
or minimum point?

00:06:52.670 --> 00:06:57.890
And to think about that, let's
imagine this point right over

00:06:57.890 --> 00:06:58.520
here.

00:06:58.520 --> 00:07:01.590
So let's call this x sub 3.

00:07:01.590 --> 00:07:03.690
If we look at the tangent
line right over here,

00:07:03.690 --> 00:07:05.390
if we look at the
slope right over here,

00:07:05.390 --> 00:07:10.300
it looks like f prime of
x sub 3 is equal to 0.

00:07:10.300 --> 00:07:12.290
So based on our definition
of critical point,

00:07:12.290 --> 00:07:16.010
x sub 3 would also
be a critical point.

00:07:16.010 --> 00:07:20.150
But it does not appear to be
a minimum or a maximum point.

00:07:20.150 --> 00:07:24.070
So a minimum or maximum
point that's not an endpoint,

00:07:24.070 --> 00:07:25.960
it's definitely going
to be a critical point.

00:07:25.960 --> 00:07:28.800
But being a critical
point by itself

00:07:28.800 --> 00:07:31.720
does not mean you're at a
minimum or maximum point.

00:07:31.720 --> 00:07:33.870
So just to be clear
that all of these points

00:07:33.870 --> 00:07:35.990
were at a minimum
or maximum point.

00:07:35.990 --> 00:07:38.527
This were at a critical
point, all of these

00:07:38.527 --> 00:07:39.360
are critical points.

00:07:39.360 --> 00:07:41.860
But this is not a
minimum or maximum point.

00:07:41.860 --> 00:07:43.810
In the next video, we'll
start to think about

00:07:43.810 --> 00:07:47.390
how you can differentiate,
or how you can tell,

00:07:47.390 --> 00:07:52.480
whether you have a minimum or
maximum at a critical point.


Implicit derivative of e^(xy^2) = x - y

WEBVTT
Kind: captions
Language: en

00:00:00.690 --> 00:00:03.180
So let's take another
implicit derivative

00:00:03.180 --> 00:00:05.380
of the somewhat
crazy relationship.

00:00:05.380 --> 00:00:07.160
And I've graphed the
relationship here.

00:00:07.160 --> 00:00:09.860
As you can see, it is
actually quite bizarre.

00:00:09.860 --> 00:00:14.300
e to the x times y squared
is equal to x minus y.

00:00:14.300 --> 00:00:17.830
This is some at least in the
range that's depicted here,

00:00:17.830 --> 00:00:21.510
the x's and the y's that
satisfy this relationship.

00:00:21.510 --> 00:00:23.890
Well let's take the
derivative of both sides.

00:00:23.890 --> 00:00:28.634
So we'll apply our derivative
operator to both sides.

00:00:28.634 --> 00:00:30.050
And actually this
is a good chance

00:00:30.050 --> 00:00:32.700
to maybe explore some
different notation.

00:00:32.700 --> 00:00:34.480
We tend to be using
that notation,

00:00:34.480 --> 00:00:37.600
but oftentimes you'll see
a derivative operator that

00:00:37.600 --> 00:00:40.270
just looks like a big
capital D. So maybe we'll

00:00:40.270 --> 00:00:42.181
do that right over here.

00:00:42.181 --> 00:00:43.180
So let me make it clear.

00:00:43.180 --> 00:00:48.770
This is the equivalent
of saying d over dx.

00:00:48.770 --> 00:00:50.990
I'll just use this
big capital D operator

00:00:50.990 --> 00:00:54.980
so that you are familiar
with the notation.

00:00:54.980 --> 00:00:58.950
And instead of using dy
dx for the derivative of y

00:00:58.950 --> 00:01:02.084
with respect to x, I'm just
going to write that as y prime.

00:01:02.084 --> 00:01:03.500
So we get a little
bit of practice

00:01:03.500 --> 00:01:05.209
with different notation.

00:01:05.209 --> 00:01:07.990
So let's take the derivative
of this thing right over here.

00:01:07.990 --> 00:01:09.699
Well we're going to
apply the chain rule.

00:01:09.699 --> 00:01:11.573
Actually, we're going
to apply the chain rule

00:01:11.573 --> 00:01:12.410
multiple times here.

00:01:12.410 --> 00:01:15.500
The derivative of e to
the something with respect

00:01:15.500 --> 00:01:20.810
to that something
is going to be e

00:01:20.810 --> 00:01:27.150
to the something
times the derivative

00:01:27.150 --> 00:01:29.920
of that something
with respect to x.

00:01:29.920 --> 00:01:34.470
So times the derivative
of xy squared.

00:01:34.470 --> 00:01:35.810
So that's our left-hand side.

00:01:35.810 --> 00:01:38.050
We aren't done taking
the derivative yet.

00:01:38.050 --> 00:01:41.900
And on our right-hand side,
the derivative of x is just 1.

00:01:41.900 --> 00:01:44.580
And the derivative
with respect to x of y

00:01:44.580 --> 00:01:49.660
is just going to be minus-- or
I could write-- negative dy dx.

00:01:49.660 --> 00:01:54.820
But instead of writing dy dx,
I'm going to write y prime.

00:01:54.820 --> 00:01:59.090
As you can tell, I like this
notation and this notation

00:01:59.090 --> 00:02:01.050
more because it
makes it explicit

00:02:01.050 --> 00:02:03.585
that I'm taking the
derivative with respect to x.

00:02:03.585 --> 00:02:05.210
Here, we just have
to assume that we're

00:02:05.210 --> 00:02:06.990
taking the derivative
with respect to x.

00:02:06.990 --> 00:02:09.440
Here, we have to assume
that's the derivative of y

00:02:09.440 --> 00:02:10.810
with respect to x.

00:02:10.810 --> 00:02:15.570
But anyway let's stick with
this notation right over here.

00:02:15.570 --> 00:02:23.190
Actually, let me make
all of my y primes,

00:02:23.190 --> 00:02:24.940
all my derivatives of
y with respect to x,

00:02:24.940 --> 00:02:28.070
let me make them pink
so I keep track of them.

00:02:28.070 --> 00:02:29.650
So once again,
this is going to be

00:02:29.650 --> 00:02:36.050
equal to e to the xy squared
times the derivative of this.

00:02:36.050 --> 00:02:37.717
Well the derivative
of this, we can just

00:02:37.717 --> 00:02:40.175
use the product and actually
a little bit of the chain rule

00:02:40.175 --> 00:02:40.690
here.

00:02:40.690 --> 00:02:45.640
So the derivative of x is just
1 times the second function.

00:02:45.640 --> 00:02:48.290
So it's going to
be times y squared.

00:02:48.290 --> 00:02:50.800
And then to that,
we're going to add

00:02:50.800 --> 00:02:53.210
the product of the
first function which

00:02:53.210 --> 00:02:59.040
is this x times the derivative
of y squared with respect to x.

00:02:59.040 --> 00:03:01.840
Well that's going to be
the derivative of y squared

00:03:01.840 --> 00:03:06.000
with respect to y, which
is just going to be 2y

00:03:06.000 --> 00:03:07.910
times the derivative
of y with respect

00:03:07.910 --> 00:03:12.605
to x, which we are now
writing as y prime.

00:03:15.200 --> 00:03:18.840
And then that's going to be
equal to 1 minus y prime.

00:03:21.930 --> 00:03:24.120
And like we've
been doing, we now

00:03:24.120 --> 00:03:26.920
have to just solve for y prime.

00:03:26.920 --> 00:03:30.850
So let's distribute
this exponential, this e

00:03:30.850 --> 00:03:33.230
to the xy squared.

00:03:33.230 --> 00:03:41.950
And we get e, or maybe I
should say y squared times e

00:03:41.950 --> 00:03:44.240
to the xy squared.

00:03:44.240 --> 00:03:45.420
So that's that.

00:03:45.420 --> 00:03:53.980
Plus 2xye to the xy squared.

00:03:53.980 --> 00:03:57.550
y prime, the derivative
of y with respect to x,

00:03:57.550 --> 00:04:04.690
is equal to 1 minus the
derivative of y with respect

00:04:04.690 --> 00:04:05.840
to x.

00:04:05.840 --> 00:04:08.530
Now let's get all of our
y primes on one side.

00:04:08.530 --> 00:04:10.680
So let's add y
prime to both sides.

00:04:10.680 --> 00:04:12.520
So let's add-- and
just to be clear,

00:04:12.520 --> 00:04:15.620
I'm adding one y
prime to both sides.

00:04:15.620 --> 00:04:18.170
So let's add a y
prime to both sides.

00:04:18.170 --> 00:04:20.860
And let's subtract this
business from both sides.

00:04:20.860 --> 00:04:28.810
So let's subtract y
squared e to the xy squared

00:04:28.810 --> 00:04:30.220
subtracting from both sides.

00:04:30.220 --> 00:04:34.880
So we're going to subtract y
squared e to the xy squared.

00:04:34.880 --> 00:04:45.750
And we are left
with 2xye to the xy

00:04:45.750 --> 00:04:51.040
squared plus 1 times y prime.

00:04:51.040 --> 00:04:52.590
We had this many
y primes and then

00:04:52.590 --> 00:04:56.480
we add another 1y prime so we
have this many plus 1 y primes.

00:04:56.480 --> 00:05:00.930
That's going to be equal
to-- well, I purposely

00:05:00.930 --> 00:05:06.530
added y prime to both sides
and so we are left with 1

00:05:06.530 --> 00:05:10.890
minus-- this is kind of a crazy
expression-- y squared times e

00:05:10.890 --> 00:05:13.050
to the xy squared.

00:05:13.050 --> 00:05:16.470
And now we just have to
divide both sides by this.

00:05:16.470 --> 00:05:19.100
And we're left with the
derivative of y with respect

00:05:19.100 --> 00:05:22.860
to x is equal to this, which
I will just copy and paste.

00:05:22.860 --> 00:05:24.250
Actually, let me
just rewrite it.

00:05:28.829 --> 00:05:29.870
Scroll down a little bit.

00:05:29.870 --> 00:05:34.680
It's equal to 1
minus y squared e

00:05:34.680 --> 00:05:40.620
to the xy squared
over this business.

00:05:40.620 --> 00:05:42.540
Let me get some more space.

00:05:42.540 --> 00:05:50.160
2xye to the xy squared plus 1.

00:05:50.160 --> 00:05:51.090
And we're done.

00:05:51.090 --> 00:05:53.950
It was kind of crazy, but
fundamentally no different

00:05:53.950 --> 00:05:58.060
than what we've been doing
in the last few examples.


Worked example: Evaluating derivative with implicit differentiation

WEBVTT
Kind: captions
Language: en

00:00:00.610 --> 00:00:04.690
We've been doing a lot
of examples where we just

00:00:04.690 --> 00:00:07.850
take implicit
derivatives, but we

00:00:07.850 --> 00:00:10.030
haven't been calculating
the actual slope

00:00:10.030 --> 00:00:12.869
of the tangent line
at a given point.

00:00:12.869 --> 00:00:14.660
And that's what I want
to do in this video.

00:00:14.660 --> 00:00:17.620
So what I want to
do is figure out

00:00:17.620 --> 00:00:24.770
the slope at x is equal to 1.

00:00:24.770 --> 00:00:26.880
So when x is equal to 1.

00:00:26.880 --> 00:00:29.000
And as you can imagine,
once we implicitly

00:00:29.000 --> 00:00:30.440
take the derivative
of this, we're

00:00:30.440 --> 00:00:34.350
going to have that as
a function of x and y.

00:00:34.350 --> 00:00:36.140
So it'll be useful
to know what y

00:00:36.140 --> 00:00:38.612
value we get to when
our x is equal to 1.

00:00:38.612 --> 00:00:40.070
So let's figure
that out right now.

00:00:40.070 --> 00:00:43.420
So when x is equal to 1, our
relationship right over here

00:00:43.420 --> 00:00:46.234
becomes 1 squared,
which is just 1

00:00:46.234 --> 00:00:52.080
plus y minus 1 to the
third power is equal to 28.

00:00:52.080 --> 00:00:53.770
Subtract 1 from both sides.

00:00:53.770 --> 00:00:58.020
You get y minus 1 to the
third power is equal to 27.

00:00:58.020 --> 00:01:00.760
It looks like the numbers
work out quite neatly for us.

00:01:00.760 --> 00:01:02.500
Take the cube root
of both sides.

00:01:02.500 --> 00:01:05.760
You get y minus 1 is equal to 3.

00:01:05.760 --> 00:01:07.110
Add 1 to both sides.

00:01:07.110 --> 00:01:08.432
You get y is equal to 4.

00:01:08.432 --> 00:01:10.140
So we really want to
figure out the slope

00:01:10.140 --> 00:01:16.620
at the point 1 comma 1 comma
4, which is right over here.

00:01:16.620 --> 00:01:19.970
When x is 1, y is 4.

00:01:19.970 --> 00:01:23.480
So we want to figure out the
slope of the tangent line

00:01:23.480 --> 00:01:24.920
right over there.

00:01:24.920 --> 00:01:28.910
So let's start doing some
implicit differentiation.

00:01:28.910 --> 00:01:31.030
So we're going to
take the derivative

00:01:31.030 --> 00:01:34.610
of both sides of this
relationship, or this equation,

00:01:34.610 --> 00:01:37.090
depending on how
you want to view it.

00:01:37.090 --> 00:01:40.640
And so let's skip down
here past the orange.

00:01:40.640 --> 00:01:43.890
So the derivative with
respect to x of x squared

00:01:43.890 --> 00:01:47.660
is going to be 2x.

00:01:47.660 --> 00:01:51.270
And then the derivative with
respect to x of something

00:01:51.270 --> 00:01:54.870
to the third power is
going to be 3 times

00:01:54.870 --> 00:02:02.060
that something squared times
the derivative of that something

00:02:02.060 --> 00:02:03.450
with respect to x.

00:02:03.450 --> 00:02:06.260
And so what's the derivative
of this with respect to x?

00:02:06.260 --> 00:02:11.200
Well the derivative of y with
respect to x is just dy dx.

00:02:11.200 --> 00:02:14.080
And then the derivative of x
with respect to x is just 1.

00:02:14.080 --> 00:02:15.720
So we have minus 1.

00:02:15.720 --> 00:02:17.670
And on the right-hand
side we just get 0.

00:02:17.670 --> 00:02:22.020
Derivative of a constant
is just equal to 0.

00:02:22.020 --> 00:02:25.420
And now we need to
solve for dy dx.

00:02:25.420 --> 00:02:27.620
So we get 2x.

00:02:27.620 --> 00:02:31.936
And so if we distribute this
business times the dy dx

00:02:31.936 --> 00:02:36.879
and times the negative 1, when
we multiply it times dy dx,

00:02:36.879 --> 00:02:39.170
we get-- and actually I'm
going to write it over here--

00:02:39.170 --> 00:02:49.347
so we get plus 3 times y
minus x squared times dy dx.

00:02:49.347 --> 00:02:51.430
And then when we multiply
it times the negative 1,

00:02:51.430 --> 00:03:01.490
we get negative 3 times y
minus y minus x squared.

00:03:01.490 --> 00:03:05.460
And then of course, all of
that is going to be equal to 0.

00:03:05.460 --> 00:03:07.530
Now all we have
to do is take this

00:03:07.530 --> 00:03:09.660
and put it on the
right-hand side.

00:03:09.660 --> 00:03:12.780
So we'll subtract it from
both sides of this equation.

00:03:12.780 --> 00:03:15.189
So on the left-hand side--
and actually all the stuff

00:03:15.189 --> 00:03:16.730
that's not a dy dx
I'm going to write

00:03:16.730 --> 00:03:18.188
in green-- so on
the left-hand side

00:03:18.188 --> 00:03:26.100
we're just left with 3 times
y minus x squared times dy dx,

00:03:26.100 --> 00:03:27.860
the derivative of
y with respect to x

00:03:27.860 --> 00:03:30.920
is equal to-- I'm just going to
subtract this from both sides--

00:03:30.920 --> 00:03:34.600
is equal to negative
2x plus this.

00:03:34.600 --> 00:03:43.090
So I could write it as 3 times
y minus x squared minus 2x.

00:03:43.090 --> 00:03:44.819
So we're adding
this to both sides

00:03:44.819 --> 00:03:46.610
and we're subtracting
this from both sides.

00:03:46.610 --> 00:03:48.160
Minus 2x.

00:03:48.160 --> 00:03:50.700
And then to solve
for dy dx, we've

00:03:50.700 --> 00:03:52.530
done this multiple
times already.

00:03:52.530 --> 00:03:56.280
To solve for the derivative
of y with respect to x.

00:03:56.280 --> 00:03:59.230
The derivative of
y with respect to x

00:03:59.230 --> 00:04:06.180
is going to be equal to 3 times
y minus x squared minus 2x.

00:04:06.180 --> 00:04:13.370
All of that over this stuff,
3 times y minus x squared.

00:04:13.370 --> 00:04:15.320
And we can leave it
just like that for now.

00:04:15.320 --> 00:04:18.240
So what is the derivative
of y with respect to x?

00:04:18.240 --> 00:04:21.120
What is the slope of the
tangent line when x is 1

00:04:21.120 --> 00:04:22.720
and y is equal to 4?

00:04:22.720 --> 00:04:25.860
Well we just have to substitute
x is equal to 1 and y

00:04:25.860 --> 00:04:28.160
equals 4 into this expression.

00:04:28.160 --> 00:04:36.840
So it's going to be equal
to 3 times 4 minus 1 squared

00:04:36.840 --> 00:04:39.900
minus 2 times 1.

00:04:39.900 --> 00:04:45.480
All of that over 3
times 4 minus 1 squared,

00:04:45.480 --> 00:04:48.485
which is equal to
4 minus 1 is 3.

00:04:48.485 --> 00:04:49.460
You square it.

00:04:49.460 --> 00:04:51.436
You get 9.

00:04:51.436 --> 00:04:53.950
9 times 3 is 27.

00:04:53.950 --> 00:04:57.490
You get 27 minus 2
in the numerator,

00:04:57.490 --> 00:05:00.110
which is going to
be equal to 25.

00:05:00.110 --> 00:05:04.900
And in the denominator, you
get 3 times 9, which is 27.

00:05:04.900 --> 00:05:07.770
So the slope is 25/27.

00:05:07.770 --> 00:05:10.102
So it's almost 1, but not quite.

00:05:10.102 --> 00:05:12.310
And that's actually what it
looks like on this graph.

00:05:12.310 --> 00:05:13.500
And actually just
to make sure you

00:05:13.500 --> 00:05:14.860
know where I got this graph.

00:05:14.860 --> 00:05:16.495
This was from Wolfram Alpha.

00:05:20.310 --> 00:05:22.540
I should have told you
that from the beginning.

00:05:22.540 --> 00:05:25.410
Anyway, hopefully
you enjoyed that.


Implicit derivative of y = cos(5x - 3y)

WEBVTT
Kind: captions
Language: en

00:00:00.450 --> 00:00:01.870
Let's say we have
the relationship

00:00:01.870 --> 00:00:05.460
y is equal to cosine
of 5x minus 3y.

00:00:05.460 --> 00:00:07.760
And what I want to find
is the rate at which y

00:00:07.760 --> 00:00:09.080
is changing with respect to x.

00:00:09.080 --> 00:00:11.390
And we'll assume that
y is a function of x.

00:00:11.390 --> 00:00:13.930
So let's do what we've
always been doing.

00:00:13.930 --> 00:00:15.970
Let's apply the
derivative operator

00:00:15.970 --> 00:00:20.460
to both sides of this equation.

00:00:20.460 --> 00:00:22.150
On the left-hand
side, right over here,

00:00:22.150 --> 00:00:28.350
we get dy/dx is equal to-- now
here on the right hand side,

00:00:28.350 --> 00:00:29.970
we're going to apply
the chain rule.

00:00:29.970 --> 00:00:32.659
The derivative of the
cosine of something,

00:00:32.659 --> 00:00:35.580
with respect to
that something, is

00:00:35.580 --> 00:00:42.390
going to be equal to negative
sine of that something.

00:00:42.390 --> 00:00:45.925
So negative sine of 5x minus 3y.

00:00:45.925 --> 00:00:47.300
And then we have
to multiply that

00:00:47.300 --> 00:00:50.590
by the derivative of that
something with respect to x.

00:00:50.590 --> 00:00:53.180
So what's the derivative of the
something with respect to x?

00:00:53.180 --> 00:00:56.340
Well the derivative of
5x with respect to x

00:00:56.340 --> 00:00:58.990
is just equal to 5.

00:00:58.990 --> 00:01:02.600
And the derivative of
negative 3y with respect to x

00:01:02.600 --> 00:01:08.120
is just negative 3 times dy/dx.

00:01:08.120 --> 00:01:12.150
Negative 3 times the derivative
of y with respect to x.

00:01:12.150 --> 00:01:14.376
And now we just need
to solve for dy/dx.

00:01:14.376 --> 00:01:17.000
And as you can see, with some of
these implicit differentiation

00:01:17.000 --> 00:01:19.080
problems, this is the hard part.

00:01:19.080 --> 00:01:22.160
And actually, let me make
that dy/dx the same color.

00:01:22.160 --> 00:01:24.210
So that we can keep
track of it easier.

00:01:24.210 --> 00:01:28.290
So this is going to be dy/dx.

00:01:28.290 --> 00:01:30.710
And then I can close
the parentheses.

00:01:30.710 --> 00:01:31.870
So how can we do it?

00:01:31.870 --> 00:01:35.350
It's just going to be a little
bit of algebra to work through.

00:01:35.350 --> 00:01:39.252
Well, we can distribute
the sine of 5x minus 3y.

00:01:39.252 --> 00:01:40.460
So let me rewrite everything.

00:01:40.460 --> 00:01:44.210
We get dy-- whoops, I'm going
to do that in the yellow color--

00:01:44.210 --> 00:01:49.180
we get dy/dx is equal
to-- you distribute

00:01:49.180 --> 00:01:51.710
the negative sine
of 5x minus 3y.

00:01:51.710 --> 00:01:54.002
You get-- so let me make sure
we know what we're doing.

00:01:54.002 --> 00:01:56.043
It's going to be, we're
going to distribute that,

00:01:56.043 --> 00:01:57.620
and we're going to
distribute that.

00:01:57.620 --> 00:01:59.500
So you're going to have
5 times all of this.

00:01:59.500 --> 00:02:04.330
So you're going to have
negative this 5 times

00:02:04.330 --> 00:02:10.380
the sine of 5x minus 3y.

00:02:10.380 --> 00:02:12.880
And then you're going to have
the negative times a negative,

00:02:12.880 --> 00:02:15.338
those are going to, you're
going to end up with a positive.

00:02:15.338 --> 00:02:19.940
And so you're going to
end up with plus 3 times

00:02:19.940 --> 00:02:26.990
the sine of 5x minus 3y dy/dx.

00:02:33.620 --> 00:02:37.070
Now what we can do
is subtract 3 sine

00:02:37.070 --> 00:02:40.160
of 5x minus 3y from both sides.

00:02:40.160 --> 00:02:44.930
So just to be clear, this
is essentially a 1 dy/dx.

00:02:44.930 --> 00:02:48.980
So if we subtract this from
both sides, we are left with--

00:02:48.980 --> 00:02:51.740
So on the left-hand side,
we're going to have a 1 dy/dx,

00:02:51.740 --> 00:02:55.460
and we're going to subtract
from that 3 sine of 5x minus 3y

00:02:55.460 --> 00:02:56.910
dy/dx's.

00:02:56.910 --> 00:03:00.850
So you're going to
have 1 minus 3--

00:03:00.850 --> 00:03:03.140
I'll keep the color
for the 3 for fun--

00:03:03.140 --> 00:03:14.910
3 sine of 5x minus 3y dy/dx's
on the left-hand side,

00:03:14.910 --> 00:03:19.170
is going to be
equal to, well, we

00:03:19.170 --> 00:03:20.864
subtracted this from both sides.

00:03:20.864 --> 00:03:23.030
So on the right-hand side,
this is going to go away.

00:03:23.030 --> 00:03:27.290
So we're just going to be
left with a negative 5 sine

00:03:27.290 --> 00:03:32.324
of 5x minus 3y.

00:03:32.324 --> 00:03:33.740
And we're in the
home stretch now.

00:03:33.740 --> 00:03:36.290
To solve for dy/dx,
we just have to divide

00:03:36.290 --> 00:03:38.390
both sides of the
equation by this.

00:03:38.390 --> 00:03:45.340
And we are left with dy/dx is
equal to this thing, negative

00:03:45.340 --> 00:03:51.870
5 times the sine of 5x minus 3y.

00:03:51.870 --> 00:04:05.520
All of that over 1 minus
3 sine of 5x minus 3y.

00:04:05.520 --> 00:04:07.650
And we are done.


Showing explicit and implicit differentiation give same result

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.140
What I want to show
you in this video

00:00:02.140 --> 00:00:04.800
is that implicit
differentiation will give you

00:00:04.800 --> 00:00:07.050
the same result as,
I guess we can say,

00:00:07.050 --> 00:00:08.620
explicit
differentiation when you

00:00:08.620 --> 00:00:10.822
can differentiate explicitly.

00:00:10.822 --> 00:00:12.530
So let's say that I
have the relationship

00:00:12.530 --> 00:00:17.300
x times the square root
of y is equal to 1.

00:00:17.300 --> 00:00:19.140
This one is actually
pretty straightforward

00:00:19.140 --> 00:00:23.620
to define explicitly in
terms of x, to solve for y.

00:00:23.620 --> 00:00:28.390
So if we divide both sides
by x, we get square root of y

00:00:28.390 --> 00:00:30.590
is equal to 1/x.

00:00:30.590 --> 00:00:34.990
And then if you square both
sides, you get y is equal to 1

00:00:34.990 --> 00:00:38.870
over x squared, which
is the same thing as x

00:00:38.870 --> 00:00:40.667
to the negative 2 power.

00:00:40.667 --> 00:00:42.250
And so if you want
the derivative of y

00:00:42.250 --> 00:00:44.550
with respect to x, this
is pretty straightforward.

00:00:44.550 --> 00:00:46.500
This is just an application
of the chain rule.

00:00:46.500 --> 00:00:51.780
So we get dy dx is
equal to negative 2 x

00:00:51.780 --> 00:00:55.630
to the negative 2 minus 1--
x to the negative 3 power.

00:00:55.630 --> 00:00:57.005
So that's pretty
straightforward.

00:00:57.005 --> 00:01:00.110
But what I want to see is if
we get the same exact result

00:01:00.110 --> 00:01:02.710
when we differentiate
implicitly.

00:01:02.710 --> 00:01:07.560
So let's apply our
derivative operator

00:01:07.560 --> 00:01:13.557
to both sides of this equation.

00:01:13.557 --> 00:01:15.140
And so let me make
it clear what we're

00:01:15.140 --> 00:01:19.157
doing-- x times the square root
of y and 1 right over there.

00:01:19.157 --> 00:01:20.740
When you apply the
derivative operator

00:01:20.740 --> 00:01:24.752
to the expression on
the left-hand side,

00:01:24.752 --> 00:01:26.960
well, actually, we're going
to apply both the product

00:01:26.960 --> 00:01:28.070
rule and the chain rule.

00:01:28.070 --> 00:01:29.910
The product rule
tells us-- so we

00:01:29.910 --> 00:01:32.100
have the product of
two functions of x.

00:01:32.100 --> 00:01:33.246
You could view it that way.

00:01:33.246 --> 00:01:34.870
So this, the product
rule tells us this

00:01:34.870 --> 00:01:37.470
is going to be the
derivative with respect

00:01:37.470 --> 00:01:50.620
to x of x times the
square root of y plus x,

00:01:50.620 --> 00:01:55.040
not taking its derivative,
times the derivative

00:01:55.040 --> 00:02:00.012
with respect to x of
the square root of y.

00:02:00.012 --> 00:02:01.470
Let me make it
clear, this bracket.

00:02:06.757 --> 00:02:08.590
And on the right-hand
side, right over here,

00:02:08.590 --> 00:02:10.850
the derivative with respect
to x of this constant,

00:02:10.850 --> 00:02:14.930
that's just going
to be equal to 0.

00:02:14.930 --> 00:02:17.040
So what does this simplify to?

00:02:17.040 --> 00:02:22.730
Well, the derivative with
respect to x of x is just 1.

00:02:22.730 --> 00:02:24.790
This simplifies to
1, so we're just

00:02:24.790 --> 00:02:28.370
going to be left with the square
root of y right over here.

00:02:28.370 --> 00:02:33.550
So this is going to simplify
to a square root of y.

00:02:33.550 --> 00:02:36.600
And what does this
over here simplify to?

00:02:36.600 --> 00:02:40.880
Well the derivative with respect
to x of the square root of y,

00:02:40.880 --> 00:02:42.590
here we want to
apply the chain rule.

00:02:42.590 --> 00:02:43.590
So let me make it clear.

00:02:43.590 --> 00:02:50.070
So we have plus this x plus
whatever business this is.

00:02:50.070 --> 00:02:53.757
And I'm going to
do this in blue.

00:02:53.757 --> 00:02:56.090
Well, it's going to be the
derivative of the square root

00:02:56.090 --> 00:02:58.645
of something with respect
to that something.

00:02:58.645 --> 00:03:00.270
Well, the derivative
of the square root

00:03:00.270 --> 00:03:02.103
of something with respect
to that something,

00:03:02.103 --> 00:03:04.160
or the derivative of
something to the 1/2

00:03:04.160 --> 00:03:09.370
with respect to that something,
is going to be 1/2 times

00:03:09.370 --> 00:03:12.920
that something to the
negative 1/2 power.

00:03:12.920 --> 00:03:16.150
Once again, this right
over here is the derivative

00:03:16.150 --> 00:03:23.089
of the square root of
y with respect to y.

00:03:23.089 --> 00:03:24.380
We've seen this multiple times.

00:03:24.380 --> 00:03:29.090
If I were to say the derivative
of the square root of x

00:03:29.090 --> 00:03:37.010
with respect to x, you would
get 1/2 x to the negative 1/2.

00:03:37.010 --> 00:03:38.872
Now I'm just doing it with y's.

00:03:38.872 --> 00:03:39.830
But we're not done yet.

00:03:39.830 --> 00:03:41.205
Remember, our
derivative operator

00:03:41.205 --> 00:03:42.580
wasn't to say with respect to y.

00:03:42.580 --> 00:03:43.550
It's with respect to x.

00:03:43.550 --> 00:03:45.640
So this only gets us
with respect to y.

00:03:45.640 --> 00:03:47.280
We need to apply the
entire chain rule.

00:03:47.280 --> 00:03:53.390
We have to multiply that
times the derivative of y

00:03:53.390 --> 00:03:56.490
with respect to x in order
to get the real derivative

00:03:56.490 --> 00:03:59.470
of this expression
with respect to x.

00:03:59.470 --> 00:04:02.040
So let's multiply
times the derivative

00:04:02.040 --> 00:04:03.175
of y with respect to x.

00:04:03.175 --> 00:04:04.300
We don't know what that is.

00:04:04.300 --> 00:04:06.130
That's actually what
we're trying to solve for.

00:04:06.130 --> 00:04:07.340
But to use the
chain rule, we just

00:04:07.340 --> 00:04:09.100
have say it's the derivative
of the square root of y

00:04:09.100 --> 00:04:10.891
with respect to y times
the derivative of y

00:04:10.891 --> 00:04:12.200
with respect to x.

00:04:12.200 --> 00:04:15.990
This is the derivative of
this thing with respect to x.

00:04:15.990 --> 00:04:18.089
So we get this on
the left-hand side.

00:04:18.089 --> 00:04:22.100
On the right-hand
side, we just have a 0.

00:04:22.100 --> 00:04:24.670
And now, once again,
we can attempt

00:04:24.670 --> 00:04:28.560
to solve for the derivative
of y with respect to x.

00:04:28.560 --> 00:04:30.250
And maybe the
easiest first step is

00:04:30.250 --> 00:04:33.470
to subtract the square
root of y from both sides

00:04:33.470 --> 00:04:34.640
of this equation.

00:04:34.640 --> 00:04:37.420
And actually, let me move
all of this stuff over,

00:04:37.420 --> 00:04:40.620
so I have, once again,
more room to work with.

00:04:40.620 --> 00:04:44.990
So let me cut it, actually.

00:04:44.990 --> 00:04:46.840
And then let me paste it.

00:04:46.840 --> 00:04:49.620
Let me move it over,
right over here.

00:04:49.620 --> 00:04:51.110
So we went from there to there.

00:04:51.110 --> 00:04:53.080
I didn't gain a
lot of real estate,

00:04:53.080 --> 00:04:54.850
but hopefully this
helps a little bit.

00:04:54.850 --> 00:04:57.320
And actually, I
don't even like that.

00:04:57.320 --> 00:04:59.280
Let me leave it
where it was before.

00:04:59.280 --> 00:05:02.160
So then, if we subtract
the square root of y

00:05:02.160 --> 00:05:05.370
from both sides-- and
I'll try to simplify

00:05:05.370 --> 00:05:13.570
as I go-- we get this thing,
which I can rewrite as x

00:05:13.570 --> 00:05:17.540
times-- well, it's just going
to be x in the numerator divided

00:05:17.540 --> 00:05:25.100
by 2 times the square root
of y. y to the negative 1/2

00:05:25.100 --> 00:05:27.250
is just the square root
of y in the denominator.

00:05:27.250 --> 00:05:29.180
And 1/2, I just put the
2 in the denominator

00:05:29.180 --> 00:05:34.080
there-- times dy dx
times the derivative of y

00:05:34.080 --> 00:05:37.090
with respect to x
is going to be equal

00:05:37.090 --> 00:05:39.840
to the negative
square root of y.

00:05:39.840 --> 00:05:43.902
I just subtracted the square
root of y from both sides.

00:05:43.902 --> 00:05:46.110
And actually, this is
something that I might actually

00:05:46.110 --> 00:05:48.330
want to copy and paste up here.

00:05:48.330 --> 00:05:51.750
So copy and then paste.

00:05:51.750 --> 00:05:56.470
So let's go back up here, just
to continue our simplification

00:05:56.470 --> 00:05:58.470
solving for dy dx.

00:05:58.470 --> 00:06:00.250
Well, to solve
for dy dx, we just

00:06:00.250 --> 00:06:04.850
have to divide both sides by
x over 2 times the square root

00:06:04.850 --> 00:06:05.690
of y.

00:06:05.690 --> 00:06:15.044
So we're left with dy dx--
or dividing both sides

00:06:15.044 --> 00:06:16.710
by this is the same
thing as multiplying

00:06:16.710 --> 00:06:21.550
by the reciprocal of
this-- is equal to 2

00:06:21.550 --> 00:06:27.220
times the square root of y
over x-- over my yellow x--

00:06:27.220 --> 00:06:32.056
times the negative
square root of y.

00:06:32.056 --> 00:06:33.680
Well, what's this
going to simplify to?

00:06:33.680 --> 00:06:35.644
This is going to be
equal to y times--

00:06:35.644 --> 00:06:38.060
the square root of y times the
square root of y is just y.

00:06:38.060 --> 00:06:40.590
The negative times the
2, you get negative 2.

00:06:40.590 --> 00:06:48.020
So you get negative 2y over x
is equal to the derivative of y

00:06:48.020 --> 00:06:49.660
with respect to x.

00:06:49.660 --> 00:06:51.350
Now you might be
saying, look, we just

00:06:51.350 --> 00:06:53.260
figured out the
derivative implicitly,

00:06:53.260 --> 00:06:58.390
and it looks very different than
what we did right over here.

00:06:58.390 --> 00:07:01.390
When we just used the power
rule, we got negative 2 x

00:07:01.390 --> 00:07:02.730
to the negative third power.

00:07:05.480 --> 00:07:08.260
The key here is to realize
that this thing right over here

00:07:08.260 --> 00:07:11.270
we could solve explicitly
in terms of-- we

00:07:11.270 --> 00:07:13.270
could solve for y.

00:07:13.270 --> 00:07:14.940
So we could just make
this substitution

00:07:14.940 --> 00:07:18.230
back here to see that these
are the exact same thing.

00:07:18.230 --> 00:07:21.830
So if we make the substitution
y is equal to 1 over x squared,

00:07:21.830 --> 00:07:25.250
you would get dy dx, the
derivative of y with respect

00:07:25.250 --> 00:07:35.040
to x, is equal to our negative
2 times 1 over x squared,

00:07:35.040 --> 00:07:38.820
and then all of that over x,
which is equal to negative 2

00:07:38.820 --> 00:07:42.720
over x to the third, which is
exactly what we have over here,

00:07:42.720 --> 00:07:46.731
negative 2 x to the
negative third power.


Implicit differentiation

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:03.010
So we've got the
equation x squared

00:00:03.010 --> 00:00:04.975
plus y squared is equal to 1.

00:00:04.975 --> 00:00:07.120
I guess we could call
it a relationship.

00:00:07.120 --> 00:00:10.200
And if we were to graph
all of the points x and y

00:00:10.200 --> 00:00:12.220
that satisfied
this relationship,

00:00:12.220 --> 00:00:15.884
we get a unit circle like this.

00:00:15.884 --> 00:00:17.550
And what I'm curious
about in this video

00:00:17.550 --> 00:00:20.170
is how we can figure out the
slope of the tangent line

00:00:20.170 --> 00:00:23.260
at any point of
this unit circle.

00:00:23.260 --> 00:00:26.350
And what immediately might be
jumping out in your brain is,

00:00:26.350 --> 00:00:29.710
well a circle defined this
way, this isn't a function.

00:00:29.710 --> 00:00:32.890
It's not y explicitly
defined as a function of x.

00:00:32.890 --> 00:00:37.140
For any x value we actually
have two possible y's

00:00:37.140 --> 00:00:40.800
that satisfy this
relationship right over here.

00:00:40.800 --> 00:00:43.480
So you might be tempted
to maybe split this up

00:00:43.480 --> 00:00:46.450
into two separate
functions of x.

00:00:46.450 --> 00:00:51.530
You could say y is equal to
the positive square root of 1

00:00:51.530 --> 00:00:52.970
minus x squared.

00:00:52.970 --> 00:00:57.550
And you could say y is equal to
the negative square root of 1

00:00:57.550 --> 00:00:59.020
minus x squared.

00:00:59.020 --> 00:01:01.530
Take the derivatives of
each of these separately.

00:01:01.530 --> 00:01:05.317
And you would be able to find
the derivative for any x,

00:01:05.317 --> 00:01:07.400
or the derivative of the
slope of the tangent line

00:01:07.400 --> 00:01:08.622
at any point.

00:01:08.622 --> 00:01:10.080
But what I want to
do in this video

00:01:10.080 --> 00:01:12.970
is literally leverage
the chain rule

00:01:12.970 --> 00:01:15.060
to take the
derivative implicitly.

00:01:15.060 --> 00:01:17.850
So that I don't have
to explicitly define y

00:01:17.850 --> 00:01:20.434
is a function of x either way.

00:01:20.434 --> 00:01:22.100
And the way we do
that is literally just

00:01:22.100 --> 00:01:26.360
apply the derivative operator
to both sides of this equation.

00:01:26.360 --> 00:01:30.100
And then apply what we
know about the chain rule.

00:01:30.100 --> 00:01:32.970
Because we are not
explicitly defining y

00:01:32.970 --> 00:01:35.790
as a function of x, and
explicitly getting y

00:01:35.790 --> 00:01:38.130
is equal to f prime
of x, they call

00:01:38.130 --> 00:01:41.450
this-- which is really just an
application of the chain rule--

00:01:41.450 --> 00:01:43.185
we call it implicit
differentiation.

00:01:47.610 --> 00:01:50.440
And what I want you to keep
in the back of your mind

00:01:50.440 --> 00:01:52.730
the entire time
is that it's just

00:01:52.730 --> 00:01:54.440
an application of
the chain rule.

00:01:54.440 --> 00:01:57.710
So let's apply the derivative
operator to both sides of this.

00:01:57.710 --> 00:01:59.210
So it's the derivative
with respect

00:01:59.210 --> 00:02:08.220
to x of x squared plus y
squared, on the left hand

00:02:08.220 --> 00:02:10.270
side of our equation.

00:02:10.270 --> 00:02:13.020
And then that's going to
be equal to the derivative

00:02:13.020 --> 00:02:15.790
with respect to x on
the right hand side.

00:02:15.790 --> 00:02:17.810
I'm just doing the
same exact thing

00:02:17.810 --> 00:02:22.190
to both sides of this equation.

00:02:22.190 --> 00:02:25.660
Now if I take the derivative
of the sum of two terms,

00:02:25.660 --> 00:02:28.090
that's the same thing as taking
the sum of the derivative.

00:02:28.090 --> 00:02:29.631
So this is going to
be the same thing

00:02:29.631 --> 00:02:33.860
as the derivative with
respect to x of x squared,

00:02:33.860 --> 00:02:38.180
plus the derivative with
respect to x of y squared.

00:02:38.180 --> 00:02:40.329
I'm writing all my
orange stuff first.

00:02:40.329 --> 00:02:40.870
So let's see.

00:02:40.870 --> 00:02:43.914
This is going to be x squared,
this is going to be y squared.

00:02:43.914 --> 00:02:46.080
And then this is going to
be equal to the derivative

00:02:46.080 --> 00:02:47.660
with respect to x of a constant.

00:02:47.660 --> 00:02:49.650
This isn't changing
with respect to x.

00:02:49.650 --> 00:02:52.530
So we just get 0.

00:02:52.530 --> 00:02:54.260
Now this first term
right over here,

00:02:54.260 --> 00:02:56.720
we have done many, many,
many, many, many, many times.

00:02:56.720 --> 00:02:58.674
The derivative with
respect to x of x squared

00:02:58.674 --> 00:02:59.840
is just the power rule here.

00:02:59.840 --> 00:03:03.030
It's going to be 2 times
x to the first power.

00:03:03.030 --> 00:03:04.930
We could just say 2x.

00:03:04.930 --> 00:03:08.690
Now what's interesting is what
we're doing right over here.

00:03:08.690 --> 00:03:12.900
The derivative with
respect to x of y squared.

00:03:12.900 --> 00:03:17.209
And the realization here is
to just apply the chain rule.

00:03:17.209 --> 00:03:19.000
If we're taking the
derivative with respect

00:03:19.000 --> 00:03:22.410
to x of this
something, we just have

00:03:22.410 --> 00:03:24.930
to take the derivative--
let me make it clear--

00:03:24.930 --> 00:03:27.690
we're just going to take the
derivative of our something.

00:03:27.690 --> 00:03:31.380
The derivative of y squared--
that's what we're taking,

00:03:31.380 --> 00:03:36.270
you can kind of view that as
a function-- with respect to y

00:03:36.270 --> 00:03:41.760
and then multiply that
times the derivative of y

00:03:41.760 --> 00:03:43.600
with respect to x.

00:03:43.600 --> 00:03:47.170
We're assuming that y does
change with respect to x. y

00:03:47.170 --> 00:03:49.020
is not some type of
a constant that we're

00:03:49.020 --> 00:03:51.260
writing just an abstract terms.

00:03:51.260 --> 00:03:53.530
So we're taking the
derivative of this whole thing

00:03:53.530 --> 00:03:56.530
with respect to y.

00:03:56.530 --> 00:03:58.780
Once again, just the chain rule.

00:03:58.780 --> 00:04:01.210
And then we're taking
the derivative of y

00:04:01.210 --> 00:04:04.479
with respect to x.

00:04:04.479 --> 00:04:06.770
It might be a little bit
clearer if you kind of thought

00:04:06.770 --> 00:04:13.560
of it as the derivative
with respect to x of y,

00:04:13.560 --> 00:04:14.590
as a function of x.

00:04:17.860 --> 00:04:21.647
This might be, or y is
a function of x squared,

00:04:21.647 --> 00:04:24.230
which is essentially another way
of writing what we have here.

00:04:24.230 --> 00:04:25.820
This might be a
little bit clearer

00:04:25.820 --> 00:04:27.640
in terms of the chain rule.

00:04:27.640 --> 00:04:29.840
The derivative of
y is a function

00:04:29.840 --> 00:04:33.000
of x squared with
respect to y of x.

00:04:33.000 --> 00:04:37.140
So the derivative of
something squared with respect

00:04:37.140 --> 00:04:40.740
to that something, times the
derivative of that something,

00:04:40.740 --> 00:04:42.030
with respect to x.

00:04:42.030 --> 00:04:43.580
This is just the chain rule.

00:04:43.580 --> 00:04:46.770
I want to say it
over and over again.

00:04:46.770 --> 00:04:49.860
This is just the chain rule.

00:04:49.860 --> 00:04:50.690
So let's do that.

00:04:50.690 --> 00:04:52.710
What do we get on the
right hand side over here?

00:04:52.710 --> 00:04:54.420
And I'll write it
over here as well.

00:04:54.420 --> 00:04:56.860
This would be equal to the
derivative of y squared

00:04:56.860 --> 00:05:03.030
with respect to y, is just
going to be 2 times y.

00:05:03.030 --> 00:05:05.690
2 times y, just an
application of the chain rule.

00:05:05.690 --> 00:05:07.550
And the derivative of
y with respect to x?

00:05:07.550 --> 00:05:08.955
Well, we don't
know what that is.

00:05:08.955 --> 00:05:11.330
So we're just going to leave
that as times the derivative

00:05:11.330 --> 00:05:14.974
of y with respect to x.

00:05:14.974 --> 00:05:16.640
So let's just write
this down over here.

00:05:16.640 --> 00:05:25.510
So we have is 2x plus the
derivative of something

00:05:25.510 --> 00:05:28.230
squared, with respect to
that something, is 2 times

00:05:28.230 --> 00:05:29.160
the something.

00:05:29.160 --> 00:05:32.310
In this case, the something
is y, so 2 times y.

00:05:32.310 --> 00:05:38.410
And then times the derivative
of y with respect to x.

00:05:38.410 --> 00:05:43.030
And this is all going
to be equal to 0.

00:05:43.030 --> 00:05:44.720
Now that was interesting.

00:05:44.720 --> 00:05:47.490
Now we have an equation
that has the derivative

00:05:47.490 --> 00:05:49.567
of a y with respect to x in it.

00:05:49.567 --> 00:05:51.650
And this is what we
essentially want to solve for.

00:05:51.650 --> 00:05:55.180
This is the slope of the
tangent line at any point.

00:05:55.180 --> 00:05:57.370
So all we have to do
at this point is solve

00:05:57.370 --> 00:05:59.720
for the derivative of
y with respect to x.

00:05:59.720 --> 00:06:01.360
Solve this equation.

00:06:01.360 --> 00:06:02.310
So let's do that.

00:06:02.310 --> 00:06:04.900
And actually just so we
can do this whole thing

00:06:04.900 --> 00:06:08.860
on the same page so we can see
where we started, let me copy

00:06:08.860 --> 00:06:11.070
and paste this up here.

00:06:11.070 --> 00:06:12.730
This is where we left off.

00:06:12.730 --> 00:06:14.620
And let's continue there.

00:06:14.620 --> 00:06:17.540
So let's say let's subtract
2x from both sides.

00:06:17.540 --> 00:06:22.220
So we're left with 2y
times the derivative of y,

00:06:22.220 --> 00:06:25.920
with respect to x, is equal
to-- we're subtracting 2x

00:06:25.920 --> 00:06:30.021
from both sides-- so it's
equal to negative 2x.

00:06:30.021 --> 00:06:32.270
And if we really want to
solve for the derivative of y

00:06:32.270 --> 00:06:35.050
with respect to x, we can
just divide both sides by 2y.

00:06:39.810 --> 00:06:42.840
And we're left with
the derivative of y

00:06:42.840 --> 00:06:43.930
with respect to x.

00:06:43.930 --> 00:06:45.340
Let's scroll down a little bit.

00:06:45.340 --> 00:06:47.540
The derivative of
y with respect to x

00:06:47.540 --> 00:06:51.930
is equal to, well
the 2s cancel out.

00:06:51.930 --> 00:06:59.070
We we're left with
negative x over y.

00:06:59.070 --> 00:07:00.410
So this is interesting.

00:07:00.410 --> 00:07:02.980
We didn't have to us
explicitly define y

00:07:02.980 --> 00:07:04.550
as a function of x here.

00:07:04.550 --> 00:07:08.710
But we got our derivative
in terms of an x and a y.

00:07:08.710 --> 00:07:10.740
Not just only in terms of an x.

00:07:10.740 --> 00:07:12.190
But what does this mean?

00:07:12.190 --> 00:07:14.660
Well, if we wanted
to find, let's

00:07:14.660 --> 00:07:18.980
say we wanted to find the
derivative at this point

00:07:18.980 --> 00:07:20.172
right over here.

00:07:20.172 --> 00:07:22.130
Which, if you're familiar
with the unit circle,

00:07:22.130 --> 00:07:25.950
so if this was a
45 degree angle,

00:07:25.950 --> 00:07:28.610
this would be the square
root of 2 over 2 comma

00:07:28.610 --> 00:07:31.090
the square root of 2 over 2.

00:07:31.090 --> 00:07:33.830
What is the slope of
the tangent line there?

00:07:33.830 --> 00:07:34.830
Well, we figured it out.

00:07:34.830 --> 00:07:38.480
It's going to be
negative x over y.

00:07:38.480 --> 00:07:46.985
So the slope of the
tangent line here

00:07:46.985 --> 00:07:49.050
is going to be
equal to negative x.

00:07:49.050 --> 00:07:53.930
So negative square root
of 2 over 2 over y.

00:07:53.930 --> 00:07:57.990
Over square root of 2 over 2,
which is equal to negative 1.

00:07:57.990 --> 00:08:01.580
And that looks just about right.


Derivative of e_____cos(e_)

WEBVTT
Kind: captions
Language: en

00:00:00.830 --> 00:00:03.300
Let's now use what we know about the chain
rule and

00:00:03.300 --> 00:00:08.020
the product rule to take the derivative of
an even weirder expression.

00:00:08.020 --> 00:00:13.206
So, we're gonna take the derivative, we're
gonna take the derivative of

00:00:13.206 --> 00:00:17.090
either the cosine of x times the cosine of
e to the x.

00:00:17.090 --> 00:00:18.880
So, let's take the derivative of this.

00:00:18.880 --> 00:00:21.360
So, we can view this as the product of two
functions.

00:00:21.360 --> 00:00:26.582
So, the product rule tells us that this is
going to be

00:00:26.582 --> 00:00:32.039
the derivative with respect to x of e to
the cosine of x,

00:00:32.039 --> 00:00:37.495
e to the cosine of x times cosine times
cosine of e to the x

00:00:37.495 --> 00:00:43.960
plus, plus the first function, just e to
the cosine of x.

00:00:43.960 --> 00:00:48.240
E to the cosine of x, times the derivative
of the second function.

00:00:48.240 --> 00:00:53.816
Times the derivative with respect to x of
cosine of e to the x.

00:00:53.816 --> 00:00:55.030
Cosine of e to the x.

00:00:59.060 --> 00:01:01.710
And so, we just need to figure out what
these two derivatives are.

00:01:01.710 --> 00:01:04.810
And so you can imagine the chain rule
might be applicable here.

00:01:04.810 --> 00:01:05.670
So, let me make it clear.

00:01:05.670 --> 00:01:08.780
This, we got from the product rule.

00:01:08.780 --> 00:01:12.380
Product, product rule.

00:01:12.380 --> 00:01:13.864
But then, to evaluate each of these

00:01:13.864 --> 00:01:15.750
derivatives, we need to use the chain
rule.

00:01:15.750 --> 00:01:17.700
So, let's think about this a little bit.

00:01:17.700 --> 00:01:22.050
So, the derivative, let me copy and paste
this so I don't have to rewrite it.

00:01:22.050 --> 00:01:25.840
So, copy and paste.

00:01:25.840 --> 00:01:29.458
So, let's think about what the derivative
of e to

00:01:29.458 --> 00:01:33.080
the cosine of x is, e to the cosine of x.

00:01:33.080 --> 00:01:36.957
So, we can view our outer function as e to
the something

00:01:36.957 --> 00:01:40.582
,as e to the something and the derivative
of e to the

00:01:40.582 --> 00:01:44.713
something with respect to something is
just going to be e to

00:01:44.713 --> 00:01:48.710
that something so it's going to be e to
the cosine of x.

00:01:48.710 --> 00:01:51.070
So, let me do that in that same blue
color.

00:01:51.070 --> 00:01:53.021
So, it's going to be e, actually I'm gonna
do it

00:01:53.021 --> 00:01:55.900
in that, actually I'm gonna do it in a new
color.

00:01:55.900 --> 00:01:57.350
Let me do it in magenta.

00:01:57.350 --> 00:02:00.222
So, the derivative of e to the something
with

00:02:00.222 --> 00:02:03.550
respect to something is just e to the
something.

00:02:03.550 --> 00:02:06.053
It's just e to the cosine of x, and we
have to

00:02:06.053 --> 00:02:10.160
multiply that times the derivative of the
something with respect to x.

00:02:10.160 --> 00:02:14.250
So, what's the derivative of cosine of x
with respect to x?

00:02:14.250 --> 00:02:16.450
Well, that's just negative sine of x.

00:02:16.450 --> 00:02:21.160
So, it's times negative sine of x.

00:02:21.160 --> 00:02:23.400
And so, we figured out this first
derivative.

00:02:23.400 --> 00:02:24.710
Let me make it clear.

00:02:24.710 --> 00:02:31.710
This right over here is the derivative of
e to the cosine of x.

00:02:31.710 --> 00:02:37.850
Derivative of e to the cosine of x, with
respect to, with respect to cosine of x.

00:02:37.850 --> 00:02:41.524
And this right over here, this right over
here is the

00:02:41.524 --> 00:02:45.710
derivative of cosine of x with respect,
with respect to x.

00:02:45.710 --> 00:02:46.940
And we just took the product of the two.

00:02:46.940 --> 00:02:49.130
That's what the chain rule tells us.

00:02:49.130 --> 00:02:50.040
Fair enough.

00:02:50.040 --> 00:02:52.850
Now, let's figure out this derivative out
here.

00:02:52.850 --> 00:02:57.360
So, we want to find the derivative with
respect to x of cosine e to the x.

00:02:57.360 --> 00:03:00.310
So, once again, let me copy and paste it.

00:03:00.310 --> 00:03:03.540
So, we need to figure out this thing right
over here.

00:03:03.540 --> 00:03:07.410
So first, just like we did, we're just
going to apply the chain rule again.

00:03:07.410 --> 00:03:11.650
We need to figure out the derivative of
cosine of something,

00:03:11.650 --> 00:03:14.860
in this case, e to the x, with respect to
that something.

00:03:14.860 --> 00:03:18.859
So, this is going to be equal to
derivative of cosine of something

00:03:18.859 --> 00:03:23.930
with respect to that something is equal to
the negative sine of that something.

00:03:23.930 --> 00:03:27.640
Negative sine of e to x, of e to the x.

00:03:27.640 --> 00:03:32.606
Once again, we can view this as the
derivative of cosine

00:03:32.606 --> 00:03:38.110
of e to the x, with respect to, with
respect to e to the x.

00:03:38.110 --> 00:03:39.863
And then we multiply that times the

00:03:39.863 --> 00:03:42.880
derivative of the something with respect
to x.

00:03:42.880 --> 00:03:46.140
So, let me do this, and this, and I'm
running out of colors.

00:03:46.140 --> 00:03:48.530
Let me do this in the screen color.

00:03:48.530 --> 00:03:52.810
So, times the derivative of e to the x
with respect to x is just e to the x.

00:03:52.810 --> 00:03:56.736
So that right over there is the derivative
of

00:03:56.736 --> 00:04:00.565
e to the x with respect, with respect to
x.

00:04:00.565 --> 00:04:03.421
And so, we're essentially done, we just
have to substitute what

00:04:03.421 --> 00:04:06.228
we found using the chain rule back into
our original expression.

00:04:06.228 --> 00:04:09.929
The derivative of this business up here is
going to be equal to, let

00:04:09.929 --> 00:04:15.410
me just copy and paste everything just to
make everything nice, nice and clean.

00:04:15.410 --> 00:04:18.880
So, copy and paste.

00:04:18.880 --> 00:04:23.004
So, that is going to be equal to, is going
to be equal to

00:04:23.004 --> 00:04:27.846
this times cosine e to the x, so this is
going to be, we'll see,

00:04:27.846 --> 00:04:31.701
we could put the e to the x out front, we
could put the

00:04:31.701 --> 00:04:37.640
negative out front, so we could write it
as negative e to the cosine x.

00:04:37.640 --> 00:04:44.762
e to the Cosine x, times Sine of x, times
Sine of x,

00:04:44.762 --> 00:04:52.110
times Cosine of e to the x, times Cosine
of e to the x.

00:04:52.110 --> 00:04:54.100
So, that's this first term here.

00:04:54.100 --> 00:05:00.740
Plus, plus e of the cosine x times all of
this stuff.

00:05:00.740 --> 00:05:03.780
And so, let's see, we can put the negative
out front again.

00:05:03.780 --> 00:05:05.350
So let's put that negative out front.

00:05:05.350 --> 00:05:10.970
So we have a negative, negative of e to
the cosine of x times e to the x.

00:05:10.970 --> 00:05:12.520
I can write it this way.

00:05:12.520 --> 00:05:16.343
e to the x times e to the cosine of x, and
you could simplify that, or combine

00:05:16.343 --> 00:05:19.199
it, since you're multiplying two things
with the same

00:05:19.199 --> 00:05:22.060
base, but I'll just leave it like this,
like this.

00:05:22.060 --> 00:05:26.500
e to the x, times e to the cosine of x,
times the sine.

00:05:26.500 --> 00:05:29.920
We already have the negative, so then we
have sine of e to the x.

00:05:29.920 --> 00:05:33.390
Sine of e to the x, right over here.

00:05:33.390 --> 00:05:37.311
Times sin, sin, of e to the x, we had
negative sin e

00:05:37.311 --> 00:05:40.867
to the x times e to the x, negative sin of
e to the

00:05:40.867 --> 00:05:44.606
x times e to the x, and then that
multiplied by e to

00:05:44.606 --> 00:05:49.810
the cosine of x so we have the exact same
thing right over here.

00:05:49.810 --> 00:05:50.270
And we're done.


Change of base formula proof

WEBVTT
Kind: captions
Language: en

00:00:00.850 --> 00:00:02.340
What I want to do
in this video is

00:00:02.340 --> 00:00:11.080
prove the change of base
formula for logarithms,

00:00:11.080 --> 00:00:16.090
which tells us-- let me
write this-- formula.

00:00:16.090 --> 00:00:21.040
Which tells us that if I want
to figure out the logarithm base

00:00:21.040 --> 00:00:28.150
a of x, that I can
figure this out

00:00:28.150 --> 00:00:30.430
by taking logarithms
with a different base.

00:00:30.430 --> 00:00:36.790
That this would be equal
to the logarithm base

00:00:36.790 --> 00:00:47.020
b-- so some other base-- base b
of x, divided by the logarithm

00:00:47.020 --> 00:00:51.720
base b of a.

00:00:54.350 --> 00:00:56.320
And this is a really
useful result.

00:00:56.320 --> 00:00:58.590
If your calculator only
has natural logarithm

00:00:58.590 --> 00:01:01.450
or log base 10, you can
now use this to figure out

00:01:01.450 --> 00:01:02.850
the logarithm using any base.

00:01:02.850 --> 00:01:04.440
If you want to figure
out the log base

00:01:04.440 --> 00:01:05.481
2-- let me make it clear.

00:01:05.481 --> 00:01:09.270
If you want to figure
out the logarithm base,

00:01:09.270 --> 00:01:14.950
let's say, base 3
of, let's say, 25,

00:01:14.950 --> 00:01:17.740
you can use your calculator
either using log base 10

00:01:17.740 --> 00:01:19.332
or log base 2.

00:01:19.332 --> 00:01:20.790
So you could say
that this is going

00:01:20.790 --> 00:01:27.810
to be equal to log
base 10 of 25--

00:01:27.810 --> 00:01:30.920
and most calculators have
a button for that-- divided

00:01:30.920 --> 00:01:40.760
by log base 10 of 3.

00:01:40.760 --> 00:01:43.500
So this is an application of
the change of base formula.

00:01:43.500 --> 00:01:45.770
But let's actually prove it.

00:01:45.770 --> 00:01:50.550
So let's say that
we want to-- let's

00:01:50.550 --> 00:02:01.980
set logarithm base a of x to
be equal to some new variable.

00:02:01.980 --> 00:02:05.170
Let's call that variable,
let's call that equal to y.

00:02:05.170 --> 00:02:09.139
So this right over here, we are
just setting that equal to y.

00:02:09.139 --> 00:02:12.160
Well, this is just another way
of saying that a to the y power

00:02:12.160 --> 00:02:13.740
is equal to x.

00:02:13.740 --> 00:02:24.630
So we can rewrite this as a
to the y power is equal to x.

00:02:24.630 --> 00:02:27.180
I'll write the x out
here, because I'm

00:02:27.180 --> 00:02:29.029
about to-- these two
things are equal.

00:02:29.029 --> 00:02:30.570
This is just another
way of restating

00:02:30.570 --> 00:02:32.320
what we just wrote up here.

00:02:32.320 --> 00:02:34.760
Now, let's introduce
the logarithm base b.

00:02:34.760 --> 00:02:37.240
And to introduce it, I'm
just to take log base

00:02:37.240 --> 00:02:40.220
b of both sides
of this equation.

00:02:40.220 --> 00:02:44.640
So let's take logarithm base
b of the left-hand side,

00:02:44.640 --> 00:02:50.680
and logarithm base b
of the right-hand side.

00:02:50.680 --> 00:02:53.710
Well, we know from our
logarithm properties

00:02:53.710 --> 00:02:56.110
that the logarithm of
something to a power

00:02:56.110 --> 00:02:59.350
is the exact same
thing as the power

00:02:59.350 --> 00:03:01.210
times the logarithm
of that something.

00:03:01.210 --> 00:03:03.040
So logarithm base
b of a to the y

00:03:03.040 --> 00:03:11.920
is the same thing as y times
the logarithm base b of a.

00:03:11.920 --> 00:03:14.870
So this is just a traditional
logarithm property.

00:03:14.870 --> 00:03:16.480
We prove it elsewhere.

00:03:16.480 --> 00:03:17.990
And we already know
it's going to be

00:03:17.990 --> 00:03:19.230
equal to the right-hand side.

00:03:19.230 --> 00:03:27.090
It's going to be equal
to log base b of x.

00:03:27.090 --> 00:03:29.190
And now, let's just solve for y.

00:03:29.190 --> 00:03:32.357
And this is exciting, because y
was this thing right over here.

00:03:32.357 --> 00:03:33.940
But now if we solve
for y, we're going

00:03:33.940 --> 00:03:37.540
to be solving for y in
terms of logarithm base b.

00:03:37.540 --> 00:03:39.170
To solve for y, we
just have to divide

00:03:39.170 --> 00:03:45.000
both sides of this equation
by log base b of a.

00:03:45.000 --> 00:03:52.970
So we divide by log base b
of a on the left-hand side,

00:03:52.970 --> 00:04:01.850
and we divide by log base b
of a on the right-hand side.

00:04:01.850 --> 00:04:04.480
And so on the left-hand
side, these two characters

00:04:04.480 --> 00:04:05.790
are going to cancel out.

00:04:05.790 --> 00:04:08.090
And we are left with-- and
we deserve a drum roll now--

00:04:08.090 --> 00:04:15.759
that y is equal to log base b
of x divided by log base b of a.

00:04:15.759 --> 00:04:16.550
So let me write it.

00:04:16.550 --> 00:04:18.533
Just copy and paste
this so I don't

00:04:18.533 --> 00:04:20.410
have to keep switching colors.

00:04:20.410 --> 00:04:21.800
So let me paste this.

00:04:21.800 --> 00:04:22.860
So there you have it.

00:04:22.860 --> 00:04:24.590
You have your change
of base formula.

00:04:24.590 --> 00:04:27.800
Remember, y is the same thing
as this thing right over here.

00:04:27.800 --> 00:04:29.290
y is log of a.

00:04:29.290 --> 00:04:33.800
Actually, let me make it clear.
y, which is equal to log of a,

00:04:33.800 --> 00:04:39.840
which is equal to log base
a of x-- so copy and paste--

00:04:39.840 --> 00:04:43.510
y, which is equal to this thing,
which is how we defined it

00:04:43.510 --> 00:04:47.120
right over here, y is equal
to log base a of x, we've just

00:04:47.120 --> 00:04:53.370
shown, is also equal to this, if
we write it in terms of base b.

00:04:53.370 --> 00:04:58.740
And we have our change
of base formula.


Quotient rule from product & chain rules

WEBVTT
Kind: captions
Language: en

00:00:00.670 --> 00:00:02.790
We already know that the
product rule tells us

00:00:02.790 --> 00:00:06.510
that if we have the product of
two functions-- so let's say

00:00:06.510 --> 00:00:10.200
f of x and g of x--
and we want to take

00:00:10.200 --> 00:00:15.520
the derivative of this
business, that this is just

00:00:15.520 --> 00:00:16.980
going to be equal
to the derivative

00:00:16.980 --> 00:00:20.280
of the first function,
f prime of x, times

00:00:20.280 --> 00:00:27.950
the second function, times g
of x, plus the first function,

00:00:27.950 --> 00:00:30.830
so not even taking its
derivative, so plus f

00:00:30.830 --> 00:00:33.165
of x times the derivative
of the second function.

00:00:37.220 --> 00:00:39.870
So two terms, in each term
we take the derivative of one

00:00:39.870 --> 00:00:42.370
of the functions and not the
other, and then we switch.

00:00:42.370 --> 00:00:45.340
So over here is the
derivative of f, not of g.

00:00:45.340 --> 00:00:47.522
Here it's the derivative
of g, not of f.

00:00:47.522 --> 00:00:49.230
This is hopefully a
little bit of review.

00:00:49.230 --> 00:00:50.790
This is the product rule.

00:00:50.790 --> 00:00:52.373
Now what we're
essentially going to do

00:00:52.373 --> 00:00:53.780
is reapply the
product rule to do

00:00:53.780 --> 00:00:56.754
what many of your calculus books
might call the quotient rule.

00:00:56.754 --> 00:00:58.670
I have mixed feelings
about the quotient rule.

00:00:58.670 --> 00:01:01.086
If you know it, it might make
some operations a little bit

00:01:01.086 --> 00:01:03.879
faster, but it really comes
straight out of the product

00:01:03.879 --> 00:01:04.379
rule.

00:01:04.379 --> 00:01:06.320
And I frankly always
forget the quotient rule,

00:01:06.320 --> 00:01:09.230
and I just rederive it
from the product rule.

00:01:09.230 --> 00:01:10.970
So let's see what
we're talking about.

00:01:10.970 --> 00:01:14.650
So let's imagine if we
had an expression that

00:01:14.650 --> 00:01:19.140
could be written as f
of x divided by g of x.

00:01:19.140 --> 00:01:21.990
And we want to take the
derivative of this business,

00:01:21.990 --> 00:01:26.700
the derivative of
f of x over g of x.

00:01:26.700 --> 00:01:29.610
The key realization
is to just recognize

00:01:29.610 --> 00:01:32.990
that this is the same thing
as the derivative of-- instead

00:01:32.990 --> 00:01:34.610
of writing f of
x over g of x, we

00:01:34.610 --> 00:01:40.355
could write this as f of x times
g of x to the negative 1 power.

00:01:44.162 --> 00:01:45.620
And now we can use
the product rule

00:01:45.620 --> 00:01:47.910
with a little bit
of the chain rule.

00:01:47.910 --> 00:01:50.520
What is this going
to be equal to?

00:01:50.520 --> 00:01:52.030
Well, we just use
the product rule.

00:01:52.030 --> 00:01:54.970
It's the derivative of the
first function right over here--

00:01:54.970 --> 00:01:59.880
so it's going to
be f prime of x--

00:01:59.880 --> 00:02:03.780
times just the second
function, which is just

00:02:03.780 --> 00:02:13.460
g of x to the negative 1
power plus the first function,

00:02:13.460 --> 00:02:17.960
which is just f of x,
times the derivative

00:02:17.960 --> 00:02:19.439
of the second function.

00:02:19.439 --> 00:02:21.980
And here we're going to have to
use a little bit of the chain

00:02:21.980 --> 00:02:22.640
rule.

00:02:22.640 --> 00:02:24.434
The derivative of
the outside, which

00:02:24.434 --> 00:02:25.850
we could kind of
view as something

00:02:25.850 --> 00:02:28.660
to the negative 1 power with
respect to that something,

00:02:28.660 --> 00:02:31.700
is going to be negative 1
times that something, which

00:02:31.700 --> 00:02:34.525
in this case is g of x
to the negative 2 power.

00:02:34.525 --> 00:02:36.150
And then we have to
take the derivative

00:02:36.150 --> 00:02:37.740
of the inside
function with respect

00:02:37.740 --> 00:02:41.880
to x, which is
just g prime of x.

00:02:41.880 --> 00:02:42.890
And there you have it.

00:02:42.890 --> 00:02:44.490
We have found the
derivative of this

00:02:44.490 --> 00:02:46.750
using the product rule
and the chain rule.

00:02:46.750 --> 00:02:48.260
Now, this is not
the form that you

00:02:48.260 --> 00:02:49.660
might see when
people are talking

00:02:49.660 --> 00:02:51.410
about the quotient
rule in your math book.

00:02:51.410 --> 00:02:53.620
So let's see if we can
simplify this a little bit.

00:02:53.620 --> 00:02:57.480
All of this is going to be equal
to-- we can write this term

00:02:57.480 --> 00:03:03.490
right over here as f
prime of x over g of x.

00:03:07.720 --> 00:03:10.160
And we could write
all of this as-- we

00:03:10.160 --> 00:03:12.020
could put this negative
sign out front.

00:03:12.020 --> 00:03:20.070
We have negative f of
x times g prime of x.

00:03:24.620 --> 00:03:28.555
And then all of that
over g of x squared.

00:03:28.555 --> 00:03:30.650
Let me write this a
little bit neater.

00:03:30.650 --> 00:03:33.835
All of that over g of x squared.

00:03:36.789 --> 00:03:38.830
And it still isn't in the
form that you typically

00:03:38.830 --> 00:03:40.240
see in your calculus book.

00:03:40.240 --> 00:03:42.855
To do that, we just have
to add these two fractions.

00:03:42.855 --> 00:03:44.980
So let's multiply the
numerator and the denominator

00:03:44.980 --> 00:03:47.720
here by g of x so that we have
everything in the form of g

00:03:47.720 --> 00:03:49.810
of x squared in the denominator.

00:03:49.810 --> 00:03:52.430
So if we multiply the
numerator by g of x,

00:03:52.430 --> 00:03:54.740
we'll get g of x right
over here and then

00:03:54.740 --> 00:03:57.530
the denominator will
be g of x squared.

00:03:57.530 --> 00:03:59.050
And now we're ready to add.

00:03:59.050 --> 00:04:02.450
And so we get the
derivative of f

00:04:02.450 --> 00:04:08.910
of x over g of x is equal to
the derivative of f of x times g

00:04:08.910 --> 00:04:15.460
of x minus-- not plus
anymore-- let me write it

00:04:15.460 --> 00:04:28.020
in white-- f of x
times g prime of x,

00:04:28.020 --> 00:04:34.320
all of that over g of x squared.

00:04:34.320 --> 00:04:36.410
So once again, you
can always derive this

00:04:36.410 --> 00:04:38.420
from the product rule
and the chain rule.

00:04:38.420 --> 00:04:41.150
Sometimes this might be
convenient to remember in order

00:04:41.150 --> 00:04:45.110
to work through some problems of
this form a little bit faster.

00:04:45.110 --> 00:04:48.010
And if you wanted to kind of see
the pattern between the product

00:04:48.010 --> 00:04:50.430
rule and the quotient
rule, the derivative of one

00:04:50.430 --> 00:04:53.050
function just times
the other function.

00:04:53.050 --> 00:04:55.710
And instead of
adding the derivative

00:04:55.710 --> 00:04:57.860
of the second function
times the first function,

00:04:57.860 --> 00:04:59.140
we now subtract it.

00:04:59.140 --> 00:05:02.190
And all that is over the
second function squared.

00:05:02.190 --> 00:05:05.212
Whatever was in the denominator,
it's all of that squared.

00:05:05.212 --> 00:05:06.670
So when we're taking
the derivative

00:05:06.670 --> 00:05:08.720
of the function in the
denominator up here,

00:05:08.720 --> 00:05:12.070
there's a subtraction, and then
we are also putting everything

00:05:12.070 --> 00:05:14.930
over the second
function squared.


Quotient rule for derivative of tan x

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.690
In the last video, we
saw that the quotient

00:00:02.690 --> 00:00:06.904
rule, which, once again, I have
mixed feelings about because it

00:00:06.904 --> 00:00:08.820
really comes straight
out of the product rule.

00:00:08.820 --> 00:00:11.484
If we have something in the
form f of x over g of x,

00:00:11.484 --> 00:00:12.900
then the derivative
of it could be

00:00:12.900 --> 00:00:14.730
this business right over here.

00:00:14.730 --> 00:00:16.840
So I thought I would at
least do one example where

00:00:16.840 --> 00:00:17.670
we can apply that.

00:00:17.670 --> 00:00:21.640
And can could do it to find the
derivative of something useful.

00:00:21.640 --> 00:00:24.609
So what's the
derivative with respect

00:00:24.609 --> 00:00:26.150
to x-- let me write
this a little bit

00:00:26.150 --> 00:00:32.356
neater-- the derivative with
respect to x of tangent of x?

00:00:32.356 --> 00:00:33.730
And you might say,
hey Sal, wait,

00:00:33.730 --> 00:00:35.120
I thought this was
about the quotient rule.

00:00:35.120 --> 00:00:36.578
But you just have
to remember, what

00:00:36.578 --> 00:00:39.370
is the definition
of the tangent of x?

00:00:39.370 --> 00:00:41.930
Or what is one way to
view the tangent of x?

00:00:41.930 --> 00:00:44.100
The tangent of x
is the same thing

00:00:44.100 --> 00:00:48.070
as sine of x-- let me
now color code it--

00:00:48.070 --> 00:00:56.860
is the same thing as sine
of x over cosine of x.

00:01:01.510 --> 00:01:07.800
And now it looks clear that
our expression is the ratio

00:01:07.800 --> 00:01:12.250
or it's one function
over another function.

00:01:12.250 --> 00:01:15.190
So now we can just
apply the quotient rule.

00:01:15.190 --> 00:01:17.160
So all of this
business is going to be

00:01:17.160 --> 00:01:21.020
equal to the derivative of
sine of x times cosine of x.

00:01:21.020 --> 00:01:23.420
So what's the
derivative of sine of x?

00:01:23.420 --> 00:01:25.200
Well, that's just cosine of x.

00:01:25.200 --> 00:01:28.440
So it's cosine of x is
derivative of sine of x times

00:01:28.440 --> 00:01:30.640
whatever function we
had in the denominator.

00:01:30.640 --> 00:01:37.886
So times cosine of
x minus whatever

00:01:37.886 --> 00:01:39.260
function we had
in the numerator,

00:01:39.260 --> 00:01:45.550
sine of x, times the
derivative of whatever we

00:01:45.550 --> 00:01:46.920
have in the denominator.

00:01:46.920 --> 00:01:50.010
Well what's the
derivative of cosine of x?

00:01:50.010 --> 00:01:53.860
Well the derivative of cosine
of x is negative sine of x.

00:01:53.860 --> 00:01:55.412
So we'll put the sine of x here.

00:01:55.412 --> 00:01:56.870
And it's a negative
so I could just

00:01:56.870 --> 00:01:59.150
make this right over
here a positive.

00:01:59.150 --> 00:02:03.120
And then all of
that over, whatever

00:02:03.120 --> 00:02:04.920
was in the denominator
squared, all

00:02:04.920 --> 00:02:11.570
of that over cosine
of x squared.

00:02:11.570 --> 00:02:13.580
Now what does this simplify to?

00:02:13.580 --> 00:02:15.280
Well in the numerator
right over here,

00:02:15.280 --> 00:02:17.940
we have cosine of x
times cosine of x.

00:02:17.940 --> 00:02:22.700
So all of this simplifies
to cosine squared of x.

00:02:22.700 --> 00:02:27.240
And sine of x times sine of x,
that's just sine squared of x.

00:02:27.240 --> 00:02:29.680
And what's cosine squared
of x plus sine squared of x?

00:02:29.680 --> 00:02:32.306
This is one of the most basic
trigonometric identities.

00:02:32.306 --> 00:02:34.430
It comes straight out of
the unit circle definition

00:02:34.430 --> 00:02:36.577
of trig functions.

00:02:36.577 --> 00:02:37.660
Let me write it over here.

00:02:37.660 --> 00:02:41.990
Cosine squared of
x plus sine squared

00:02:41.990 --> 00:02:47.500
of x is equal to 1, which
simplifies things quite nicely.

00:02:47.500 --> 00:02:50.750
So cosine squared of
x plus sine squared

00:02:50.750 --> 00:02:54.940
of x, all of this entire
numerator is equal to 1.

00:02:54.940 --> 00:03:03.460
So this nicely simplifies to
1 over cosine of x squared,

00:03:03.460 --> 00:03:06.110
which we could also write like
this, cosine squared of x.

00:03:06.110 --> 00:03:09.620
These are two ways of
writing cosine of x squared,

00:03:09.620 --> 00:03:15.630
which is the same exact thing
as 1 over cosine of x squared,

00:03:15.630 --> 00:03:18.010
which is the same
thing as secant.

00:03:18.010 --> 00:03:20.840
1 over cosine of x
is just secant of x.

00:03:20.840 --> 00:03:23.310
Secant of x squared, or we
could write it like this.

00:03:23.310 --> 00:03:25.654
Secant squared of x.

00:03:25.654 --> 00:03:27.070
And so that's where
it comes from.

00:03:27.070 --> 00:03:29.520
If you know that the
derivative of sine of x

00:03:29.520 --> 00:03:31.410
is cosine of x and the
derivative of cosine

00:03:31.410 --> 00:03:34.094
of x is negative sine of x,
we can use the quotient rule,

00:03:34.094 --> 00:03:36.260
which, once again, comes
straight out of the product

00:03:36.260 --> 00:03:38.260
rule to find the
derivative of tangent

00:03:38.260 --> 00:03:41.730
x is secant squared of x.


Derivative of sin(ln(x_))

WEBVTT
Kind: captions
Language: en

00:00:00.269 --> 00:00:01.810
So now we're going
to attempt to take

00:00:01.810 --> 00:00:06.720
the derivative of the sine of
the natural log of x squared.

00:00:06.720 --> 00:00:10.320
So now we have a function that's
the composite of a function,

00:00:10.320 --> 00:00:12.005
that's a composite
of another function.

00:00:12.005 --> 00:00:13.380
So one way you
could think of it,

00:00:13.380 --> 00:00:22.780
if you set f of x as being equal
to sine of x, and g of x being

00:00:22.780 --> 00:00:32.500
the natural log of x, and
h of x equaling x squared.

00:00:32.500 --> 00:00:35.580
Then this thing right over
here is the exact same thing

00:00:35.580 --> 00:00:38.340
as trying to take the
derivative with respect

00:00:38.340 --> 00:00:48.070
to x of f of g of h of x.

00:00:51.730 --> 00:00:53.410
And what I want to
do is kind of think

00:00:53.410 --> 00:00:55.030
about how I would
do it in my head,

00:00:55.030 --> 00:00:58.031
without having to write all
the chain rule notation.

00:00:58.031 --> 00:00:59.530
So the way I would
think about this,

00:00:59.530 --> 00:01:01.613
if I were doing this in
my head, is the derivative

00:01:01.613 --> 00:01:03.780
of this outer function
of f, with respect

00:01:03.780 --> 00:01:07.390
to the level of composition
directly below it.

00:01:07.390 --> 00:01:10.630
So the derivative of
sine of x is cosine of x.

00:01:10.630 --> 00:01:12.590
But instead of it
being a cosine of x,

00:01:12.590 --> 00:01:17.420
it's going to be cosine of
whatever was inside of here.

00:01:17.420 --> 00:01:19.530
So it's going to be
cosine of natural log--

00:01:19.530 --> 00:01:22.070
let me write that
in that same color--

00:01:22.070 --> 00:01:26.715
cosine of natural
log of x squared.

00:01:29.492 --> 00:01:31.200
I'm going to do x that
same yellow color.

00:01:36.210 --> 00:01:38.600
And so you could really view
this, this part what I just

00:01:38.600 --> 00:01:45.990
read over here, as f prime,
this is f prime of g of h of x.

00:01:45.990 --> 00:01:49.380
This is f prime of g of h of x.

00:01:53.371 --> 00:01:54.870
If you want to keep
track of things.

00:01:54.870 --> 00:01:57.161
So I just took the derivative
of the outer with respect

00:01:57.161 --> 00:01:58.459
to whatever was inside of it.

00:01:58.459 --> 00:02:00.000
And now I have to
take the derivative

00:02:00.000 --> 00:02:02.510
of the inside with respect to x.

00:02:02.510 --> 00:02:04.410
But now we have another
composite function.

00:02:04.410 --> 00:02:06.290
So we're going to
multiply this times, we're

00:02:06.290 --> 00:02:10.060
going to do the
chain rule again.

00:02:10.060 --> 00:02:13.130
We're going to take the
derivative of ln with respect

00:02:13.130 --> 00:02:14.600
to x squared.

00:02:14.600 --> 00:02:18.220
So the derivative
of ln of x is 1/x.

00:02:18.220 --> 00:02:20.720
But now we're going
to have 1 over not x,

00:02:20.720 --> 00:02:21.930
but 1 over x squared.

00:02:25.630 --> 00:02:32.345
So to be clear, this part right
over here is g prime of not x.

00:02:32.345 --> 00:02:34.660
If it was g prime of x,
this would be 1 over x.

00:02:34.660 --> 00:02:37.460
But instead of an x, we
have our h of x there.

00:02:37.460 --> 00:02:38.810
We have our x squared.

00:02:38.810 --> 00:02:41.750
So it's g prime of x squared.

00:02:41.750 --> 00:02:44.190
And then finally, we
can take the derivative

00:02:44.190 --> 00:02:46.280
of our inner function.

00:02:46.280 --> 00:02:47.630
Let me write it.

00:02:47.630 --> 00:02:49.790
So we could write this
is g prime of h of x.

00:02:52.126 --> 00:02:53.500
And finally, we
just have to take

00:02:53.500 --> 00:02:57.740
the derivative of our innermost
function with respect to x.

00:02:57.740 --> 00:03:00.390
So the derivative of
2x with respect to--

00:03:00.390 --> 00:03:03.910
or the derivative of x squared
with respect to x is 2x.

00:03:03.910 --> 00:03:07.010
So times h prime of x.

00:03:07.010 --> 00:03:08.800
Let me make everything clear.

00:03:08.800 --> 00:03:13.205
So what we have right
over here in purple,

00:03:13.205 --> 00:03:19.620
this, this, and this
are the same things.

00:03:19.620 --> 00:03:23.450
One expressed concretely,
one expressed abstractly.

00:03:23.450 --> 00:03:29.100
This, this, and this
are the same thing,

00:03:29.100 --> 00:03:31.720
expressed concretely
and abstractly.

00:03:31.720 --> 00:03:37.050
And then finally, this and
this are the same thing,

00:03:37.050 --> 00:03:39.237
expressed concretely
and abstractly.

00:03:39.237 --> 00:03:40.070
But then we're done.

00:03:40.070 --> 00:03:44.420
All we have to do to be done
is to just simplify this.

00:03:44.420 --> 00:03:47.100
So if we just change the order
in which we're multiplying,

00:03:47.100 --> 00:03:49.850
we have 2x over x squared.

00:03:49.850 --> 00:03:51.230
So I can cancel some out.

00:03:51.230 --> 00:03:53.820
So this x over x,
2x over x squared

00:03:53.820 --> 00:03:56.610
is the same thing as 2 over x.

00:03:56.610 --> 00:03:59.160
And we're multiplying it
times all of this business.

00:03:59.160 --> 00:04:01.000
So we're left with 2 over x.

00:04:01.000 --> 00:04:02.420
And this goes away.

00:04:02.420 --> 00:04:09.870
2 over x times the cosine of
the natural log of x squared.

00:04:09.870 --> 00:04:11.950
So it seemed like a very
daunting derivative.

00:04:11.950 --> 00:04:14.630
But we just say, OK, what's the
derivative of sine of something

00:04:14.630 --> 00:04:15.970
with respect to that something?

00:04:15.970 --> 00:04:18.040
Well, that's cosine
of that something.

00:04:18.040 --> 00:04:19.680
And then we go in
one layer, what's

00:04:19.680 --> 00:04:21.294
the derivative of
that something?

00:04:21.294 --> 00:04:23.460
Well, in that something we
have another composition.

00:04:23.460 --> 00:04:27.466
So the derivative of ln of x,
or ln of something with respect

00:04:27.466 --> 00:04:28.840
to another something,
well that's

00:04:28.840 --> 00:04:30.320
going to be 1 over
the something.

00:04:30.320 --> 00:04:32.330
So we had gotten a 1
over x squared here,

00:04:32.330 --> 00:04:34.660
that squared got canceled out.

00:04:34.660 --> 00:04:37.192
And then finally, the derivative
of this innermost function,

00:04:37.192 --> 00:04:38.650
it's kind of like
peeling an onion.

00:04:38.650 --> 00:04:40.440
The derivative of
this inner function

00:04:40.440 --> 00:04:42.482
with respect to x,
which was just 2x.

00:04:42.482 --> 00:04:43.690
Which we got right over here.

00:04:43.690 --> 00:04:44.870
This was 1 over x squared.

00:04:44.870 --> 00:04:48.160
This was 2x before we
did any canceling out.

00:04:48.160 --> 00:04:52.100
So hopefully that helps
clear things up a little bit.


Worked example: Derivative of ÃÂ(3x_-x) using the chain rule

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:03.480
What I want to do in this video
is start with the abstract--

00:00:03.480 --> 00:00:06.330
actually, let me call it
formula for the chain rule,

00:00:06.330 --> 00:00:09.760
and then learn to apply it
in the concrete setting.

00:00:09.760 --> 00:00:12.596
So let's start off
with some function,

00:00:12.596 --> 00:00:14.220
some expression that
could be expressed

00:00:14.220 --> 00:00:18.200
as the composition
of two functions.

00:00:18.200 --> 00:00:22.100
So it can be expressed
as f of g of x.

00:00:25.980 --> 00:00:28.610
So it's a function that can
be expressed as a composition

00:00:28.610 --> 00:00:30.130
or expression that
can be expressed

00:00:30.130 --> 00:00:32.890
as a composition
of two functions.

00:00:32.890 --> 00:00:35.135
Let me get that same color.

00:00:35.135 --> 00:00:36.680
I want the colors
to be accurate.

00:00:36.680 --> 00:00:39.470
And my goal is to take the
derivative of this business,

00:00:39.470 --> 00:00:42.390
the derivative
with respect to x.

00:00:42.390 --> 00:00:44.380
And what the chain
rule tells us is

00:00:44.380 --> 00:00:46.880
that this is going to be
equal to the derivative

00:00:46.880 --> 00:00:49.930
of the outer function with
respect to the inner function.

00:00:49.930 --> 00:00:54.460
And we can write that as f
prime of not x, but f prime

00:00:54.460 --> 00:00:56.730
of g of x, of the
inner function.

00:00:56.730 --> 00:01:00.950
f prime of g of x
times the derivative

00:01:00.950 --> 00:01:02.635
of the inner function
with respect to x.

00:01:05.810 --> 00:01:08.230
Now this might seem all
very abstract and math-y.

00:01:08.230 --> 00:01:09.760
How do you actually apply it?

00:01:09.760 --> 00:01:12.180
Well, let's try it
with a real example.

00:01:12.180 --> 00:01:17.246
Let's say we were trying
to take the derivative

00:01:17.246 --> 00:01:28.560
of the square root of
3x squared minus x.

00:01:28.560 --> 00:01:31.070
So how could we
define an f and a g

00:01:31.070 --> 00:01:32.710
so this really is
the composition

00:01:32.710 --> 00:01:35.340
of f of x and g of x?

00:01:35.340 --> 00:01:37.530
Well, we could define f of x.

00:01:37.530 --> 00:01:42.440
If we defined f of x as being
equal to the square root of x,

00:01:42.440 --> 00:01:51.430
and if we defined g of x as
being equal to 3x squared

00:01:51.430 --> 00:01:55.410
minus x, then what
is f of g of x?

00:01:55.410 --> 00:02:00.066
Well, f of g of x is
going to be equal to-- I'm

00:02:00.066 --> 00:02:01.940
going to try to keep
all the colors accurate,

00:02:01.940 --> 00:02:04.090
hopefully it'll help
for the understanding.

00:02:04.090 --> 00:02:08.039
f of g of x is equal to--
where everywhere you see the x,

00:02:08.039 --> 00:02:14.820
you replace with the g of x--
the principal root of g of x,

00:02:14.820 --> 00:02:19.110
which is equal to the
principal root of-- we

00:02:19.110 --> 00:02:22.750
defined g of x right over
here-- 3x squared minus x.

00:02:26.940 --> 00:02:29.020
So this thing right
over here is exactly

00:02:29.020 --> 00:02:32.340
f of g of x if we define
f of x in this way

00:02:32.340 --> 00:02:34.580
and g of x in this way.

00:02:34.580 --> 00:02:35.700
Fair enough.

00:02:35.700 --> 00:02:37.310
So let's apply the chain rule.

00:02:37.310 --> 00:02:42.020
What is f prime of g of
x going to be equal to,

00:02:42.020 --> 00:02:45.570
the derivative of f
with respect to g?

00:02:45.570 --> 00:02:47.840
Well, what's f prime of x?

00:02:47.840 --> 00:02:51.320
f prime of x is equal to--
this is the same thing

00:02:51.320 --> 00:02:55.090
as x to the 1/2 power, so we
can just apply the power rule.

00:02:55.090 --> 00:02:58.140
So it's going to be
1/2 times x to the--

00:02:58.140 --> 00:03:01.190
and then we just take 1 away
from the exponent, 1/2 minus 1

00:03:01.190 --> 00:03:03.820
is negative 1/2.

00:03:03.820 --> 00:03:07.190
And so what is f
prime of g of x?

00:03:11.860 --> 00:03:14.050
Well, wherever in the
derivative we saw an x,

00:03:14.050 --> 00:03:15.550
we can replace it with a g of x.

00:03:15.550 --> 00:03:18.320
So it's going to be
1/2 times-- instead

00:03:18.320 --> 00:03:25.070
of an x to the negative 1/2, we
can write a g of x to the 1/2.

00:03:25.070 --> 00:03:29.090
And this is just going
to be equal to-- let

00:03:29.090 --> 00:03:30.710
me write it right over here.

00:03:30.710 --> 00:03:35.830
It's going to be
equal to 1/2 times

00:03:35.830 --> 00:03:39.080
all of this business to
the negative 1/2 power.

00:03:39.080 --> 00:03:46.140
So 3x squared minus x,
which is exactly what we

00:03:46.140 --> 00:03:49.500
need to solve right over
here. f prime of g of x

00:03:49.500 --> 00:03:51.340
is equal to this.

00:03:51.340 --> 00:03:55.300
So this part right
over here I will--

00:03:55.300 --> 00:03:57.740
let me square it off in green.

00:03:57.740 --> 00:03:59.920
What we're trying to
solve right over here,

00:03:59.920 --> 00:04:02.250
f prime of g of x,
we've just figured out

00:04:02.250 --> 00:04:05.450
is exactly this thing
right over here.

00:04:05.450 --> 00:04:08.930
So the derivative of f of the
outer function with respect

00:04:08.930 --> 00:04:10.450
to the inner function.

00:04:10.450 --> 00:04:11.390
So let me write it.

00:04:11.390 --> 00:04:17.870
It is equal to 1/2 times g
of x to the negative 1/2,

00:04:17.870 --> 00:04:22.480
times 3x squared minus x.

00:04:22.480 --> 00:04:25.490
This is exactly this
based on how we've defined

00:04:25.490 --> 00:04:28.020
f of x and how we've
defined g of x.

00:04:28.020 --> 00:04:29.520
Conceptually, if
you're just looking

00:04:29.520 --> 00:04:31.311
at this, the derivative
of the outer thing,

00:04:31.311 --> 00:04:33.889
you're taking something
to the 1/2 power.

00:04:33.889 --> 00:04:35.430
So the derivative
of that whole thing

00:04:35.430 --> 00:04:37.940
with respect to your something
is going to be 1/2 times

00:04:37.940 --> 00:04:40.120
that something to the
negative 1/2 power.

00:04:40.120 --> 00:04:42.040
That's essentially
what we're saying.

00:04:42.040 --> 00:04:44.400
But now we have to take the
derivative of our something

00:04:44.400 --> 00:04:45.530
with respect to x.

00:04:48.370 --> 00:04:50.090
And that's more straightforward.

00:04:50.090 --> 00:04:53.160
g prime of x-- we just use
the power rule for each

00:04:53.160 --> 00:04:55.390
of these terms-- is
equal to 6x to the first,

00:04:55.390 --> 00:04:57.910
or just 6x minus 1.

00:04:57.910 --> 00:05:03.530
So this part right over here
is just going to be 6x minus 1.

00:05:03.530 --> 00:05:06.460
Just to be clear,
this right over here

00:05:06.460 --> 00:05:09.400
is this right over here
and we're multiplying.

00:05:09.400 --> 00:05:10.490
And we're done.

00:05:10.490 --> 00:05:12.330
We have just applied
the power rule.

00:05:12.330 --> 00:05:14.270
So just to review,
it's the derivative

00:05:14.270 --> 00:05:16.730
of the outer function
with respect to the inner.

00:05:16.730 --> 00:05:19.520
So instead of having
1/2x to the negative 1/2,

00:05:19.520 --> 00:05:22.430
it's 1/2 g of x to
the negative 1/2,

00:05:22.430 --> 00:05:24.760
times the derivative of the
inner function with respect

00:05:24.760 --> 00:05:27.100
to x, times the derivative
of g with respect

00:05:27.100 --> 00:05:30.128
to x, which is right over there.


Product rule

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.780
What we will talk
about in this video

00:00:02.780 --> 00:00:04.910
is the product
rule, which is one

00:00:04.910 --> 00:00:08.960
of the fundamental ways
of evaluating derivatives.

00:00:08.960 --> 00:00:10.570
And we won't prove
it in this video,

00:00:10.570 --> 00:00:12.700
but we will learn
how to apply it.

00:00:12.700 --> 00:00:15.860
And all it tells us is that
if we have a function that

00:00:15.860 --> 00:00:19.250
can be expressed as a product
of two functions-- so let's

00:00:19.250 --> 00:00:22.960
say it can be expressed as
f of x times g of x-- and we

00:00:22.960 --> 00:00:27.610
want to take the derivative
of this function,

00:00:27.610 --> 00:00:30.270
that it's going to be equal
to the derivative of one

00:00:30.270 --> 00:00:32.350
of these functions,
f prime of x--

00:00:32.350 --> 00:00:34.530
let's say the derivative
of the first one times

00:00:34.530 --> 00:00:40.400
the second function
plus the first function,

00:00:40.400 --> 00:00:42.900
not taking its derivative,
times the derivative

00:00:42.900 --> 00:00:44.770
of the second function.

00:00:44.770 --> 00:00:46.120
So here we have two terms.

00:00:46.120 --> 00:00:48.560
In each term, we took
the derivative of one

00:00:48.560 --> 00:00:50.220
of the functions
and not the other,

00:00:50.220 --> 00:00:52.470
and we multiplied the
derivative of the first function

00:00:52.470 --> 00:00:54.800
times the second
function plus just

00:00:54.800 --> 00:00:56.520
the first function
times the derivative

00:00:56.520 --> 00:00:58.320
of the second function.

00:00:58.320 --> 00:01:00.890
Now let's see if we can actually
apply this to actually find

00:01:00.890 --> 00:01:02.790
the derivative of something.

00:01:02.790 --> 00:01:05.700
So let's say we are dealing
with-- I don't know--

00:01:05.700 --> 00:01:12.227
let's say we're dealing with
x squared times cosine of x.

00:01:12.227 --> 00:01:13.560
Or let's say-- well, yeah, sure.

00:01:13.560 --> 00:01:15.320
Let's do x squared
times sine of x.

00:01:15.320 --> 00:01:16.960
Could have done it either way.

00:01:16.960 --> 00:01:20.340
And we are curious about
taking the derivative of this.

00:01:20.340 --> 00:01:23.910
We are curious about
what its derivative is.

00:01:23.910 --> 00:01:26.710
Well, we might
immediately recognize

00:01:26.710 --> 00:01:28.430
that this is the
product of-- this

00:01:28.430 --> 00:01:30.880
can be expressed as a
product of two functions.

00:01:30.880 --> 00:01:35.320
We could set f of x
is equal to x squared,

00:01:35.320 --> 00:01:37.840
so that is f of x
right over there.

00:01:37.840 --> 00:01:44.175
And we could set g of x
to be equal to sine of x.

00:01:44.175 --> 00:01:45.050
And there we have it.

00:01:45.050 --> 00:01:48.054
We have our f of x times g of x.

00:01:48.054 --> 00:01:50.470
And we could think about what
these individual derivatives

00:01:50.470 --> 00:01:51.140
are.

00:01:51.140 --> 00:01:54.520
The derivative of f of x is
just going to be equal to 2x

00:01:54.520 --> 00:01:59.270
by the power rule, and
the derivative of g of x

00:01:59.270 --> 00:02:00.920
is just the derivative
of sine of x,

00:02:00.920 --> 00:02:02.545
and we covered this
when we just talked

00:02:02.545 --> 00:02:03.600
about common derivatives.

00:02:03.600 --> 00:02:06.740
Derivative of sine
of x is cosine of x.

00:02:06.740 --> 00:02:09.250
And so now we're ready to
apply the product rule.

00:02:09.250 --> 00:02:13.260
This is going to be equal to
f prime of x times g of x.

00:02:13.260 --> 00:02:15.390
So f prime of x--
the derivative of f

00:02:15.390 --> 00:02:24.260
is 2x times g of x, which
is sine of x plus just

00:02:24.260 --> 00:02:28.360
our function f,
which is x squared

00:02:28.360 --> 00:02:31.810
times the derivative of
g, times cosine of x.

00:02:35.170 --> 00:02:35.860
And we're done.

00:02:35.860 --> 00:02:39.430
We just applied
the product rule.