Differentiating rational functions

WEBVTT
Kind: captions
Language: en

00:00:00.492 --> 00:00:01.325
- [Voiceover] Let's say that

00:00:01.325 --> 00:00:05.965
y is equal to five minus three
x over x squared plus three x

00:00:05.965 --> 00:00:06.798
And we want to figure out

00:00:06.798 --> 00:00:09.729
what's the derivative
of y in respect to x.

00:00:09.729 --> 00:00:12.618
Now it might immediately
jump out at you that look,

00:00:12.618 --> 00:00:16.900
y is being defined as a
rational expression here

00:00:16.900 --> 00:00:20.210
as the quotient of two
different expressions.

00:00:20.210 --> 00:00:23.009
We could even view this as
two different functions.

00:00:23.009 --> 00:00:27.296
You could view this one up here as u of x,

00:00:27.296 --> 00:00:29.452
so you could say this is the same thing,

00:00:29.452 --> 00:00:32.702
this is the same thing as u of x, over,

00:00:33.758 --> 00:00:37.407
you could view the one in
the denominator as v of x.

00:00:37.407 --> 00:00:39.807
That one right there is v of x

00:00:39.807 --> 00:00:41.479
and so if you're taking
the derivative of something

00:00:41.479 --> 00:00:43.000
that can be expressed in this way,

00:00:43.000 --> 00:00:45.868
as the quotient of two
different functions,

00:00:45.868 --> 00:00:47.933
well then you could use the quotient rule.

00:00:47.933 --> 00:00:50.022
I'll give you my little
aside, like I always do

00:00:50.022 --> 00:00:51.329
the quotient rule, if you ever forget it

00:00:51.329 --> 00:00:52.831
it can be derived from the product rule

00:00:52.831 --> 00:00:53.839
and we have videos there,

00:00:53.839 --> 00:00:55.875
cause the product rule's
a bit easier to remember.

00:00:55.875 --> 00:00:56.988
But what I can do is just say,

00:00:56.988 --> 00:01:00.914
"look, dy dx, if y is
just u of x over v of x",

00:01:00.914 --> 00:01:02.985
I'm just gonna restate the quotient rule.

00:01:02.985 --> 00:01:05.239
This is going to be, this is going to be

00:01:05.239 --> 00:01:09.828
the derivative of the
function in the numerator, so.

00:01:09.828 --> 00:01:11.078
d, dx of u of x

00:01:14.689 --> 00:01:17.003
times the function in the denominator

00:01:17.003 --> 00:01:19.836
times v of x minus, I'll do the

00:01:23.682 --> 00:01:28.071
minus the function in
the numerator, u of x,

00:01:28.071 --> 00:01:30.883
times the derivative of the
function in the denominator

00:01:30.883 --> 00:01:32.383
times d dx, v of x

00:01:36.940 --> 00:01:38.633
and we're almost there

00:01:38.633 --> 00:01:42.800
and then over, over the function
in the denominator squared

00:01:45.079 --> 00:01:48.412
the function in the denominator squared.

00:01:50.376 --> 00:01:53.299
So this might look messy
but all we have to do now

00:01:53.299 --> 00:01:56.391
is think about what is
the derivative of u of x?

00:01:56.391 --> 00:01:58.741
What is the derivative of v of x?

00:01:58.741 --> 00:02:00.642
And we should just be able
to substitute those things

00:02:00.642 --> 00:02:03.991
back into this expression
we just wrote down.

00:02:03.991 --> 00:02:05.386
So let's do that.

00:02:05.386 --> 00:02:09.553
So the derivative with respect
to x of u of x, of u of x

00:02:12.849 --> 00:02:16.651
is equal to, let's see,
five minus three x.

00:02:16.651 --> 00:02:18.958
The derivative of five is zero.

00:02:18.958 --> 00:02:20.907
The derivative of negative three x,

00:02:20.907 --> 00:02:23.094
well that's just gonna be negative three.

00:02:23.094 --> 00:02:24.581
That's just negative three.

00:02:24.581 --> 00:02:26.404
If any of that look
completely unfamiliar to you

00:02:26.404 --> 00:02:29.111
I encourage you to review
the derivative properties

00:02:29.111 --> 00:02:31.330
and maybe the power rule.

00:02:31.330 --> 00:02:33.660
Now let's think about
what is the derivative

00:02:33.660 --> 00:02:37.779
with respect to x,
derivative with respect to x,

00:02:37.779 --> 00:02:39.529
of v of x, of v of x?

00:02:41.602 --> 00:02:42.743
Well derivative of x squared,

00:02:42.743 --> 00:02:44.541
we just bring that exponent out front,

00:02:44.541 --> 00:02:47.317
it's gonna be two times
x to the two minus one

00:02:47.317 --> 00:02:51.258
or two x to the first power or just two x

00:02:51.258 --> 00:02:54.360
and then the derivative
of three x is just three.

00:02:54.360 --> 00:02:56.721
So two x plus three.

00:02:56.721 --> 00:03:00.112
Now we know everything we need
to substitute back in here.

00:03:00.112 --> 00:03:03.508
The derivative of u with respect to x,

00:03:03.508 --> 00:03:06.588
this right over here
is just negative three.

00:03:06.588 --> 00:03:10.755
V of x, this we know is
x squared plus three x.

00:03:13.413 --> 00:03:16.731
We know that this right
over here is v of x.

00:03:16.731 --> 00:03:21.233
And then u of x we know
is five minus three x.

00:03:21.233 --> 00:03:22.816
Five minus three x.

00:03:24.787 --> 00:03:27.458
The derivative of v with respect to x

00:03:27.458 --> 00:03:31.625
we know is two x plus
three, two x plus three.

00:03:33.184 --> 00:03:37.351
And then finally v of x, we
know is x squared plus three x.

00:03:38.252 --> 00:03:41.252
So this is x squared plus three x

00:03:42.217 --> 00:03:44.257
and so what do we get?

00:03:44.257 --> 00:03:48.612
Well we are going to get, it's
gonna look a little bit hairy

00:03:48.612 --> 00:03:51.445
It's going to be equal to negative

00:03:52.953 --> 00:03:57.120
I'll focus this so first we
have this business up here.

00:03:58.461 --> 00:04:00.725
Negative three times x
squared plus three x.

00:04:00.725 --> 00:04:02.754
So I'm just gonna distribute
the negative three.

00:04:02.754 --> 00:04:06.754
So it's negative three
x squared minus nine x

00:04:07.888 --> 00:04:11.191
and then from that we
are going to subtract

00:04:11.191 --> 00:04:15.663
the product of these two
expressions and so let's see,

00:04:15.663 --> 00:04:17.746
what is that going to be?

00:04:18.760 --> 00:04:21.510
Well, we have a five times two x,

00:04:24.690 --> 00:04:25.857
which is ten x

00:04:27.014 --> 00:04:29.514
a five times three which is 15

00:04:32.449 --> 00:04:35.734
we have a negative three x times two x

00:04:35.734 --> 00:04:39.734
so that is going to be
negative six x squared

00:04:40.869 --> 00:04:45.568
minus six x squared, and then
a negative three x times three

00:04:45.568 --> 00:04:47.151
so negative nine x.

00:04:48.272 --> 00:04:50.238
And let's see we can
simplify that a little bit.

00:04:50.238 --> 00:04:52.747
Ten x minus nine x,

00:04:52.747 --> 00:04:55.983
well that's just going to leave us with x.

00:04:55.983 --> 00:04:59.580
So ten x minus nine x
is just going to be x

00:04:59.580 --> 00:05:02.958
and then in our denominator,
we're almost there.

00:05:02.958 --> 00:05:05.067
In our denominator we
could just write that as

00:05:05.067 --> 00:05:08.651
x plus three x, x squared
plus three x squared

00:05:08.651 --> 00:05:10.479
or if we want we could expand it out,

00:05:10.479 --> 00:05:11.612
I'll just leave it like that

00:05:11.612 --> 00:05:14.362
x squared plus three x squared

00:05:16.304 --> 00:05:18.678
and so if we wanna simplify

00:05:18.678 --> 00:05:20.936
or attempt to simplify this a little bit.

00:05:20.936 --> 00:05:25.818
It's going to be negative
three x squared minus nine x

00:05:25.818 --> 00:05:29.568
and then you're gonna
have a negative minus x

00:05:31.084 --> 00:05:33.417
minus x and then minus 15

00:05:36.653 --> 00:05:39.529
and then minus negative six x squared

00:05:39.529 --> 00:05:43.029
so plus six x squared, all of that over

00:05:47.269 --> 00:05:50.479
x squared plus three x squared

00:05:50.479 --> 00:05:54.101
or x squared plus three x, squared.

00:05:54.101 --> 00:05:55.454
I should say it that way.

00:05:55.454 --> 00:05:59.049
Now let's see this numerator
I can simplify a little bit.

00:05:59.049 --> 00:06:01.596
Negative three x squared
plus six x squared

00:06:01.596 --> 00:06:05.429
that's going to be
positive three x squared

00:06:06.283 --> 00:06:08.533
and then we have in orange,

00:06:10.263 --> 00:06:13.810
we have negative nine x minus an x,

00:06:13.810 --> 00:06:17.560
well that's gonna be
minus ten x, minus ten x

00:06:18.671 --> 00:06:21.461
and then we have minus 15.

00:06:21.461 --> 00:06:22.461
So minus 15.

00:06:23.377 --> 00:06:26.430
So there you have it, we
finally have finished.

00:06:26.430 --> 00:06:28.847
This is all going to be equal to

00:06:28.847 --> 00:06:30.557
this is all going to be equal to

00:06:30.557 --> 00:06:34.057
three x squared minus ten x minus 15

00:06:35.531 --> 00:06:38.781
over x squared plus three x, squared

00:06:41.500 --> 00:06:42.833
and we are done.


Worked example: Quotient rule with table

WEBVTT
Kind: captions
Language: en

00:00:00.371 --> 00:00:02.254
- [Voiceover] Let f be
a function such that

00:00:02.254 --> 00:00:04.010
f of negative one is equal to three,

00:00:04.010 --> 00:00:06.367
f prime of negative one is equal to five.

00:00:06.367 --> 00:00:08.372
Let g be the function g of x is

00:00:08.372 --> 00:00:10.485
equal to two x to the third power.

00:00:10.485 --> 00:00:13.207
Let capital F be a function defined as,

00:00:13.207 --> 00:00:16.224
so capital F is defined
as lowercase f of x

00:00:16.224 --> 00:00:19.028
divided by lowercase g
of x, and they want us

00:00:19.028 --> 00:00:21.247
to evaluate the derivative of capital F

00:00:21.247 --> 00:00:23.809
at x equals negative one.

00:00:23.809 --> 00:00:25.040
So the way that we can do that is,

00:00:25.040 --> 00:00:27.672
let's just take the
derivative of capital F,

00:00:27.672 --> 00:00:30.728
and then evaluate it at x equals one.

00:00:30.728 --> 00:00:32.713
And the way they've set up capital F,

00:00:32.713 --> 00:00:34.331
this function definition, we can see that

00:00:34.331 --> 00:00:36.595
it is a quotient of two functions.

00:00:36.595 --> 00:00:38.852
So if we want to take it's derivative,

00:00:38.852 --> 00:00:40.175
you might say, well, maybe the

00:00:40.175 --> 00:00:42.419
quotient rule is important here.

00:00:42.419 --> 00:00:44.005
And I'll always give you my aside.

00:00:44.005 --> 00:00:46.110
The quotient rule, I'm
gonna state it right now,

00:00:46.110 --> 00:00:48.148
it could be useful to know it,

00:00:48.148 --> 00:00:49.470
but in case you ever forget it,

00:00:49.470 --> 00:00:51.043
you can derive it pretty quickly

00:00:51.043 --> 00:00:52.905
from the product rule, and if you know it,

00:00:52.905 --> 00:00:54.220
the chain rule combined, you can

00:00:54.220 --> 00:00:56.059
get the quotient rule pretty quick.

00:00:56.059 --> 00:00:59.130
But let me just state the
quotient rule right now.

00:00:59.130 --> 00:01:01.554
So if you have some function defined as

00:01:01.554 --> 00:01:03.303
some function in the numerator

00:01:03.303 --> 00:01:07.330
divided by some function
in the denominator,

00:01:07.330 --> 00:01:08.911
we can say its derivative, and this is

00:01:08.911 --> 00:01:11.579
really just a restatement
of the quotient rule,

00:01:11.579 --> 00:01:14.573
its derivative is going to
be the derivative of the

00:01:14.573 --> 00:01:17.573
function of the numerator, so d, dx,

00:01:19.361 --> 00:01:23.966
f of x, times the function
in the denominator,

00:01:23.966 --> 00:01:28.133
so times g of x, minus the
function in the numerator,

00:01:31.802 --> 00:01:34.591
minus f of x, not taking its derivative,

00:01:34.591 --> 00:01:38.171
times the derivative in the
function of the denominator,

00:01:38.171 --> 00:01:40.838
d, dx, g of x, all of that over,

00:01:45.657 --> 00:01:48.156
so all of this is going to be over

00:01:48.156 --> 00:01:50.513
the function in the denominator squared.

00:01:50.513 --> 00:01:54.680
So this g of x squared,
g of x, g of x squared.

00:01:56.509 --> 00:01:58.948
And you can use different
types of notation here.

00:01:58.948 --> 00:02:01.570
You could say, instead
of writing this with

00:02:01.570 --> 00:02:03.022
a derivative operator,
you could say this is

00:02:03.022 --> 00:02:06.772
the same thing as g
prime of x, and likewise,

00:02:07.702 --> 00:02:12.090
you could say, well that is
the same thing as f prime of x.

00:02:12.090 --> 00:02:14.340
And so now we just want
to evaluate this thing,

00:02:14.340 --> 00:02:16.046
and you might say, wait, how
do I evaluate this thing?

00:02:16.046 --> 00:02:17.356
Well, let's just try it.

00:02:17.356 --> 00:02:20.100
Let's just say we want to evaluate F prime

00:02:20.100 --> 00:02:22.396
when x is equal to negative one.

00:02:22.396 --> 00:02:26.563
So we can write F prime of
negative one is equal to,

00:02:27.897 --> 00:02:30.443
well everywhere we see an x,
let's put a negative one here.

00:02:30.443 --> 00:02:33.825
It's going to be f prime of negative one,

00:02:33.825 --> 00:02:36.914
lowercase f prime, that's
a little confusing,

00:02:36.914 --> 00:02:41.081
lowercase f prime of negative
one times g of negative one,

00:02:42.645 --> 00:02:46.062
g of negative one minus f of negative one

00:02:51.803 --> 00:02:54.303
times g prime of negative one.

00:02:57.708 --> 00:03:02.359
All of that over, we'll
do that in the same color,

00:03:02.359 --> 00:03:04.692
so take my color seriously.

00:03:04.692 --> 00:03:08.859
Alright, all of that over
g of negative one squared.

00:03:12.756 --> 00:03:15.252
Now can we figure out what
F prime of negative one

00:03:15.252 --> 00:03:17.075
f of negative one, g of negative one,

00:03:17.075 --> 00:03:19.721
and g prime of negative
one, what they are?

00:03:19.721 --> 00:03:21.309
Well some of them, they tell us outright.

00:03:21.309 --> 00:03:23.674
They tell us f and f
prime at negative one,

00:03:23.674 --> 00:03:26.865
and for g, we can
actually solve for those.

00:03:26.865 --> 00:03:28.766
So, let's see, if this is, let's

00:03:28.766 --> 00:03:31.804
first evaluate g of negative one.

00:03:31.804 --> 00:03:34.175
G of negative one is going to be two

00:03:34.175 --> 00:03:37.916
times negative one to the third power.

00:03:37.916 --> 00:03:40.293
Well negative one to the third
power is just negative one,

00:03:40.293 --> 00:03:43.113
times two, so this is negative two,

00:03:43.113 --> 00:03:47.113
and g prime of x, I'll
do it here, g prime of x.

00:03:48.525 --> 00:03:50.956
Let's use the power rule,
bring that three out front,

00:03:50.956 --> 00:03:55.479
three times two is six, x,
decrement that exponent,

00:03:55.479 --> 00:03:59.729
three minus one is two, and
so g prime of negative one

00:03:59.729 --> 00:04:03.572
is equal to six times
negative one squared.

00:04:03.572 --> 00:04:04.989
Well negative one squared is just one,

00:04:04.989 --> 00:04:06.764
so this is going to be equal to six.

00:04:06.764 --> 00:04:09.241
So we actually know what
all of these values are now.

00:04:09.241 --> 00:04:10.914
We know, so first we wanna figure out

00:04:10.914 --> 00:04:13.475
f prime of negative one.

00:04:13.475 --> 00:04:14.852
Well they tell us that right over here.

00:04:14.852 --> 00:04:17.928
F prime of negative one is equal to five.

00:04:17.928 --> 00:04:19.261
So that is five.

00:04:20.420 --> 00:04:24.939
G of negative one, well we
figured that right here.

00:04:24.939 --> 00:04:28.503
G of negative one is negative two.

00:04:28.503 --> 00:04:30.503
So this is negative two.

00:04:32.000 --> 00:04:35.333
F of negative one, so f of negative one,

00:04:37.190 --> 00:04:40.237
they tell us that right over there.

00:04:40.237 --> 00:04:42.472
That is equal to three.

00:04:42.472 --> 00:04:45.222
And then g prime of negative one,

00:04:47.115 --> 00:04:50.490
just circle it in this green color,

00:04:50.490 --> 00:04:52.633
g prime of negative
one, we figured it out.

00:04:52.633 --> 00:04:54.464
It is equal to six.

00:04:54.464 --> 00:04:56.662
So this is equal to six.

00:04:56.662 --> 00:05:00.059
And then finally, g of
negative one right over here.

00:05:00.059 --> 00:05:02.011
We already figured that out.

00:05:02.011 --> 00:05:04.594
That was equal to negative two.

00:05:06.475 --> 00:05:09.276
So this is all going to simplify to...

00:05:09.276 --> 00:05:12.113
So you have five times negative two,

00:05:12.113 --> 00:05:15.780
which is negative 10,
minus three times six,

00:05:16.627 --> 00:05:21.004
which is 18, all of that
over negative two squared.

00:05:21.004 --> 00:05:24.120
Well negative two squared is
just going to be positive four.

00:05:24.120 --> 00:05:25.632
So this is going to be equal to

00:05:25.632 --> 00:05:28.215
negative 28 over positive four,

00:05:29.082 --> 00:05:31.963
which is equal to negative seven.

00:05:31.963 --> 00:05:33.019
And there you have it.

00:05:33.019 --> 00:05:34.481
It looks intimidating at first,

00:05:34.481 --> 00:05:35.786
but just say, okay, look.

00:05:35.786 --> 00:05:39.175
I can use the quotient
rule right over here,

00:05:39.175 --> 00:05:40.833
and then once I apply the quotient rule,

00:05:40.833 --> 00:05:42.155
I can actually just directly figure out

00:05:42.155 --> 00:05:44.907
what g of negative one,
g prime of negative one,

00:05:44.907 --> 00:05:46.510
and they gave us f of negative one,

00:05:46.510 --> 00:05:49.874
f prime of negative one, so
hopefully you find that helpful.


Worked example: Product rule with mixed implicit & explicit

WEBVTT
Kind: captions
Language: en

00:00:00.587 --> 00:00:02.226
- [Voiceover] Let F be
a function such that

00:00:02.226 --> 00:00:03.842
F of negative one is three.

00:00:03.842 --> 00:00:07.054
And F prime of negative
one is equal to five.

00:00:07.054 --> 00:00:08.770
Let G be the function G of X

00:00:08.770 --> 00:00:10.191
is equal to one over X.

00:00:10.191 --> 00:00:12.831
Let capital F be a function defined as

00:00:12.831 --> 00:00:15.438
the product of those other two functions.

00:00:15.438 --> 00:00:19.066
What is capital F prime of negative one?

00:00:19.066 --> 00:00:21.030
Well, we can just apply
the product rule here,

00:00:21.030 --> 00:00:22.462
let me just rewrite let me just

00:00:22.462 --> 00:00:25.154
essentially state the product rule.

00:00:25.154 --> 00:00:27.365
Capital F prime of X

00:00:27.365 --> 00:00:29.619
is going to be equal to

00:00:29.619 --> 00:00:31.929
since capital F of X is the product

00:00:31.929 --> 00:00:32.834
of these two functions,

00:00:32.834 --> 00:00:34.744
when we apply the product
rule this is gonna be

00:00:34.744 --> 00:00:35.744
F prime of X

00:00:36.903 --> 00:00:37.903
times G of X

00:00:39.297 --> 00:00:40.130
plus plus

00:00:41.595 --> 00:00:43.166
F of X

00:00:43.166 --> 00:00:44.749
times G prime of X.

00:00:46.328 --> 00:00:47.882
And so if we want to evaluate this

00:00:47.882 --> 00:00:49.918
at F of negative one

00:00:49.918 --> 00:00:51.727
capital F prime

00:00:51.727 --> 00:00:53.950
at negative one is equal to

00:00:53.950 --> 00:00:56.193
F prime of negative one

00:00:56.193 --> 00:00:58.110
times G of negative one

00:00:59.297 --> 00:01:02.932
plus function F evaluated at negative one

00:01:02.932 --> 00:01:04.845
times the derivative of G

00:01:04.845 --> 00:01:06.912
evaluated at negative one.

00:01:06.912 --> 00:01:09.271
And let's see if we can
figure these things out.

00:01:09.271 --> 00:01:11.531
So do they tell us this anywhere?

00:01:11.531 --> 00:01:12.732
Can we figure this out?

00:01:12.732 --> 00:01:14.428
F prime of negative one.

00:01:14.428 --> 00:01:15.482
Well they tell us right over here

00:01:15.482 --> 00:01:16.667
F prime of negative one

00:01:16.667 --> 00:01:18.330
is equal to five.

00:01:18.330 --> 00:01:20.578
So this is equal to five.

00:01:20.578 --> 00:01:22.548
Now, let's actually stick with F.

00:01:22.548 --> 00:01:24.332
What is F of negative one?

00:01:24.332 --> 00:01:25.868
Well, they tell that to us

00:01:25.868 --> 00:01:26.701
right over here.

00:01:26.701 --> 00:01:29.164
F of negative one is equal to three.

00:01:29.164 --> 00:01:32.083
So F of negative one is equal to three.

00:01:32.083 --> 00:01:33.612
Now G of negative one

00:01:33.612 --> 00:01:35.046
and G prime of negative one

00:01:35.046 --> 00:01:36.759
they don't give it to us explicitly here

00:01:36.759 --> 00:01:38.300
but we can figure it out.

00:01:38.300 --> 00:01:41.149
We can, we know that if G of X

00:01:41.149 --> 00:01:42.651
is equal to this

00:01:42.651 --> 00:01:45.026
G of negative one is equal

00:01:45.026 --> 00:01:46.696
to one over negative one

00:01:46.696 --> 00:01:48.697
which is equal to negative one.

00:01:48.697 --> 00:01:51.080
So this is equal to

00:01:51.080 --> 00:01:52.559
negative one.

00:01:52.559 --> 00:01:55.348
And then last but not least

00:01:55.348 --> 00:01:57.910
if we want to find G prime of negative one

00:01:57.910 --> 00:02:00.084
we just have to take
the derivative of this.

00:02:00.084 --> 00:02:01.417
So G prime of X.

00:02:02.598 --> 00:02:04.860
Actually just let me rewrite G of X.

00:02:04.860 --> 00:02:05.693
G of X

00:02:06.601 --> 00:02:08.164
one over X is just the same thing

00:02:08.164 --> 00:02:10.243
as X to the negative one.

00:02:10.243 --> 00:02:11.506
So we're gonna use a power rule

00:02:11.506 --> 00:02:13.246
to figure out G prime of X

00:02:13.246 --> 00:02:15.357
is equal to bring that exponent

00:02:15.357 --> 00:02:17.755
out front, negative one times X

00:02:17.755 --> 00:02:20.120
to the and then decrement the exponent

00:02:20.120 --> 00:02:22.307
negative two power.

00:02:22.307 --> 00:02:23.474
So, G prime of

00:02:24.891 --> 00:02:26.670
negative one

00:02:26.670 --> 00:02:28.735
of negative one is equal to

00:02:28.735 --> 00:02:30.235
negative one times

00:02:31.854 --> 00:02:33.437
negative one to the

00:02:34.275 --> 00:02:36.220
negative two power.

00:02:36.220 --> 00:02:38.375
And that's just the same thing as

00:02:38.375 --> 00:02:41.625
negative one over negative one squared.

00:02:43.423 --> 00:02:44.452
This is one.

00:02:44.452 --> 00:02:46.258
So this is just all going to evaluate

00:02:46.258 --> 00:02:47.591
to negative one.

00:02:49.388 --> 00:02:50.804
So this is

00:02:50.804 --> 00:02:52.123
negative one.

00:02:52.123 --> 00:02:53.869
And so we have

00:02:53.869 --> 00:02:55.491
five times negative one

00:02:55.491 --> 00:02:57.267
which is negative five.

00:02:57.267 --> 00:02:58.889
Plus three times negative one

00:02:58.889 --> 00:03:00.532
which is negative three.

00:03:00.532 --> 00:03:01.911
Which is equal to

00:03:01.911 --> 00:03:03.859
negative eight.

00:03:03.859 --> 00:03:05.755
So F prime of negative one

00:03:05.755 --> 00:03:08.005
is equal to negative eight.


Worked example: Product rule with table

WEBVTT
Kind: captions
Language: en

00:00:00.604 --> 00:00:02.343
- [Voiceover] The following
tables lists the values

00:00:02.343 --> 00:00:04.889
of functions f and h,
and of their derivatives,

00:00:04.889 --> 00:00:07.771
f prime and h prime for
x is equal to three.

00:00:07.771 --> 00:00:08.604
So all this is telling us,

00:00:08.604 --> 00:00:09.437
with x is equal to three,

00:00:09.437 --> 00:00:12.362
the value of the function
is six, f of three is six,

00:00:12.362 --> 00:00:13.449
you could view it that way.

00:00:13.449 --> 00:00:16.963
h of three is zero, f
prime of three is six,

00:00:16.963 --> 00:00:19.313
and h prime of three is four.

00:00:19.313 --> 00:00:20.818
And now they want us to
evaluate the derivative

00:00:20.818 --> 00:00:25.270
with respect to x of the
product of f of x and h of x

00:00:25.270 --> 00:00:27.165
when x is equal to three.

00:00:27.165 --> 00:00:28.818
One way you could view this is,

00:00:28.818 --> 00:00:31.459
if we viewed some function,

00:00:31.459 --> 00:00:34.308
if we viewed some function g of x.

00:00:34.308 --> 00:00:38.439
g of x as being equal
to the product of f of x

00:00:38.439 --> 00:00:39.356
and h of x,

00:00:40.685 --> 00:00:42.945
this expression is the
derivative of g of x.

00:00:42.945 --> 00:00:47.112
So we could write g prime of
x is equal to the derivative

00:00:48.057 --> 00:00:50.060
with respect to x

00:00:50.060 --> 00:00:50.893
of f of x

00:00:52.318 --> 00:00:53.401
times h of x.

00:00:55.487 --> 00:00:56.989
Which is what we see right here,

00:00:56.989 --> 00:00:59.480
and which is what we want to
evaluate at x equals three.

00:00:59.480 --> 00:01:00.763
So we essentially want to evaluate

00:01:00.763 --> 00:01:01.596
g prime

00:01:02.761 --> 00:01:04.102
of three.

00:01:04.102 --> 00:01:06.320
This is what they're asking us to do.

00:01:06.320 --> 00:01:08.949
Well, to do that, let's go first up here.

00:01:08.949 --> 00:01:09.956
Let's just think about what this is doing.

00:01:09.956 --> 00:01:11.153
They're asking us to take the derivative

00:01:11.153 --> 00:01:13.436
with respect to x of the
product of two functions

00:01:13.436 --> 00:01:15.444
that we have some information about.

00:01:15.444 --> 00:01:17.320
Well, if we're taking the
derivative of the product

00:01:17.320 --> 00:01:18.691
of two functions, you
could imagine that the

00:01:18.691 --> 00:01:21.376
product rule could prove useful here.

00:01:21.376 --> 00:01:23.243
So I'm just gonna
restate the product rule.

00:01:23.243 --> 00:01:25.032
This is going to be
equal to the derivative

00:01:25.032 --> 00:01:26.562
of the first function,

00:01:26.562 --> 00:01:27.645
f prime of x,

00:01:28.795 --> 00:01:32.674
times the second function,
not taking its derivative.

00:01:32.674 --> 00:01:35.535
Plus the first function,
not taking its derivative,

00:01:35.535 --> 00:01:38.773
f of x, times the derivative
of the second function,

00:01:38.773 --> 00:01:39.856
h prime of x.

00:01:40.734 --> 00:01:43.022
So if you're trying to
find g prime of three,

00:01:43.022 --> 00:01:44.747
well that's just going to be

00:01:44.747 --> 00:01:47.497
f prime of three times h of three

00:01:49.314 --> 00:01:50.564
plus f of three

00:01:52.035 --> 00:01:53.952
times h prime of three.

00:01:55.416 --> 00:01:57.412
And lucky for us, they give
us what all these things

00:01:57.412 --> 00:01:58.412
evaluate to.

00:01:59.407 --> 00:02:03.008
f prime of three, right
over here, they tell us.

00:02:03.008 --> 00:02:07.091
f prime when x is equal
to three is equal to six.

00:02:08.099 --> 00:02:11.414
So this right over here is six.

00:02:11.414 --> 00:02:14.368
h of three, they give us that, too.

00:02:14.368 --> 00:02:17.782
h of three, when x is
three, the value of our

00:02:17.782 --> 00:02:19.997
function is zero.

00:02:19.997 --> 00:02:21.903
So this is zero.

00:02:21.903 --> 00:02:24.451
So this first term is you
just get six times zero,

00:02:24.451 --> 00:02:26.918
which is gonna be zero,
but we'll get to that.

00:02:26.918 --> 00:02:28.168
Now f of three.

00:02:29.447 --> 00:02:30.364
f of three.

00:02:31.471 --> 00:02:34.006
Well, the function when
x is equal to three,

00:02:34.006 --> 00:02:36.256
f of three is equal to six.

00:02:38.044 --> 00:02:39.867
So that is six.

00:02:39.867 --> 00:02:41.335
And then finally,

00:02:41.335 --> 00:02:44.052
h prime evaluated at three,

00:02:44.052 --> 00:02:46.690
h prime of x when x is equal to three,

00:02:46.690 --> 00:02:48.562
h prime of x is equal to four.

00:02:48.562 --> 00:02:52.121
Or you could say this is h prime of three.

00:02:52.121 --> 00:02:53.454
So this is four.

00:02:54.874 --> 00:02:56.148
And so there you have it.

00:02:56.148 --> 00:02:58.325
This is going to evaluate
to six times zero,

00:02:58.325 --> 00:03:00.790
which, that's all just gonna be zero,

00:03:00.790 --> 00:03:03.059
plus six times four,

00:03:03.059 --> 00:03:05.968
which is going to be equal to 24.

00:03:05.968 --> 00:03:07.218
And we're done.


Product rule example

WEBVTT
Kind: captions
Language: en

00:00:00.475 --> 00:00:01.425
- [Voiceover] So let's see if we can find

00:00:01.425 --> 00:00:02.852
the derivative with respect to x,

00:00:02.852 --> 00:00:05.056
with either x times the cosine of x.

00:00:05.056 --> 00:00:06.320
And like always, pause this video

00:00:06.320 --> 00:00:07.486
and give it a go on your own

00:00:07.486 --> 00:00:09.402
before we work through it.

00:00:09.402 --> 00:00:11.074
So when you look at this you might say,

00:00:11.074 --> 00:00:11.907
"well, I know how to find

00:00:11.907 --> 00:00:13.682
"the derivative with e to the x,"

00:00:13.682 --> 00:00:15.221
that's infact just e to the x.

00:00:15.221 --> 00:00:16.687
And let me write this down.

00:00:16.687 --> 00:00:18.372
We know a few things.

00:00:18.372 --> 00:00:22.095
We know the derivative
with respect to x of

00:00:22.095 --> 00:00:23.576
e to the x.

00:00:23.576 --> 00:00:26.510
E to the x is e to the x.

00:00:26.510 --> 00:00:29.484
We know how to find the
derivative cosine of x.

00:00:29.484 --> 00:00:33.364
The derivative with
respect to x of cosine of x

00:00:33.364 --> 00:00:36.200
is equal to negative sine of x.

00:00:36.200 --> 00:00:39.765
But, how do we find the
derivative of their product?

00:00:39.765 --> 00:00:41.745
Well as you can imagine,
this might involve

00:00:41.745 --> 00:00:42.934
the product rule.

00:00:42.934 --> 00:00:43.922
And let me just write down

00:00:43.922 --> 00:00:45.693
the product rule generally first.

00:00:45.693 --> 00:00:48.614
So if we take the
derivative with respect to x

00:00:48.614 --> 00:00:52.510
of the first expression in terms of x,

00:00:52.510 --> 00:00:56.343
so this is, we could
call this u of x times

00:00:57.881 --> 00:01:00.257
another expression that involves x.

00:01:00.257 --> 00:01:01.757
So u times v of x.

00:01:03.775 --> 00:01:05.664
This is going to be equal to,

00:01:05.664 --> 00:01:07.310
and I'm color-coding it so we can really

00:01:07.310 --> 00:01:08.508
keep track of things.

00:01:08.508 --> 00:01:11.728
This is going to be
equal to the derivative

00:01:11.728 --> 00:01:13.728
of the first expression.

00:01:15.054 --> 00:01:18.645
So I could write that as u prime of x

00:01:18.645 --> 00:01:21.650
times just the second expression
not the derivative of it,

00:01:21.650 --> 00:01:22.820
just the second expression.

00:01:22.820 --> 00:01:26.153
So times v of x and then we have plus

00:01:28.773 --> 00:01:30.792
the first expression, not its derivative,

00:01:30.792 --> 00:01:32.475
just the first expression.

00:01:32.475 --> 00:01:36.773
U of x times the derivative
of the second expression.

00:01:36.773 --> 00:01:39.405
Times the derivative of
the second expression.

00:01:39.405 --> 00:01:40.620
So the way you remember it is,

00:01:40.620 --> 00:01:42.015
you have these two things here,

00:01:42.015 --> 00:01:44.534
you're going to end up
with two different terms.

00:01:44.534 --> 00:01:46.280
In each of them, you're going to take

00:01:46.280 --> 00:01:48.274
the derivative of one of
them, but not the other one,

00:01:48.274 --> 00:01:49.649
and then the other one
you'll take the derivative of

00:01:49.649 --> 00:01:51.698
the other one, but not the first one.

00:01:51.698 --> 00:01:56.272
So, the derivative of u
times v is u prime times v,

00:01:56.272 --> 00:01:58.022
plus u times v prime.

00:01:58.889 --> 00:02:00.196
When you just look at it like that,

00:02:00.196 --> 00:02:01.395
it seems a little bit abstract and

00:02:01.395 --> 00:02:02.735
that might even be a little bit confusing,

00:02:02.735 --> 00:02:05.204
but that's why we have
a tangible example here

00:02:05.204 --> 00:02:06.996
and I color-coded intentionally.

00:02:06.996 --> 00:02:11.163
We could say that u of x
is equal to e to the x.

00:02:12.364 --> 00:02:16.112
And v of x is equal to the cosine of x.

00:02:16.112 --> 00:02:19.195
So v of x is equal to cosine of x.

00:02:20.294 --> 00:02:23.273
And if u of x is equal to e to the x,

00:02:23.273 --> 00:02:24.794
we know that the derivative of that

00:02:24.794 --> 00:02:26.678
with respect to x is still e to the x.

00:02:26.678 --> 00:02:29.154
That's one of the most
magical things in mathematics.

00:02:29.154 --> 00:02:31.602
One of the things that makes e so special.

00:02:31.602 --> 00:02:36.037
So u prime of x is still
equal to e to the x.

00:02:36.037 --> 00:02:37.454
And v prime of x,

00:02:39.826 --> 00:02:42.128
we know as negative sine of x.

00:02:42.128 --> 00:02:45.170
Negative sine of x, and so,

00:02:45.170 --> 00:02:46.681
what's this going to be equal to?

00:02:46.681 --> 00:02:48.347
This is going to be equal to

00:02:48.347 --> 00:02:50.803
the derivative of the first expression.

00:02:50.803 --> 00:02:53.664
So, the derivative of e
to the x which is just,

00:02:53.664 --> 00:02:56.687
e to the x, times the second expression,

00:02:56.687 --> 00:03:01.151
not taking it's derivative,
so times cosine of x.

00:03:01.151 --> 00:03:03.318
Plus the first expression,

00:03:04.970 --> 00:03:08.387
not taking its derivative, so e to the x,

00:03:10.197 --> 00:03:13.120
times the derivative of
the second expression.

00:03:13.120 --> 00:03:16.115
So, times the derivative of cosine of x

00:03:16.115 --> 00:03:18.032
which is negative sine.

00:03:19.022 --> 00:03:20.496
Negative sine of x.

00:03:20.496 --> 00:03:21.829
And it might be a little bit confusing,

00:03:21.829 --> 00:03:24.060
because e to the x is its own derivative.

00:03:24.060 --> 00:03:26.094
This right over here, you can view this as

00:03:26.094 --> 00:03:27.800
this was the derivative as e to the x

00:03:27.800 --> 00:03:29.156
which happens to be e to the x.

00:03:29.156 --> 00:03:31.145
That's what's exciting
about that expression,

00:03:31.145 --> 00:03:32.205
or that function.

00:03:32.205 --> 00:03:34.085
And then this is just e
to the x without taking

00:03:34.085 --> 00:03:36.549
it's derivative - they are
of course, the same thing.

00:03:36.549 --> 00:03:38.797
But anyway, now we can just simplify it.

00:03:38.797 --> 00:03:41.380
This is going to be equal to...

00:03:43.051 --> 00:03:47.218
We could write either as e
to the x times cosine of x,

00:03:48.974 --> 00:03:52.141
times cosine of x minus e to the x.

00:03:53.426 --> 00:03:55.926
E to the x times sine of x.

00:03:57.447 --> 00:03:59.384
Times sine of x.

00:03:59.384 --> 00:04:01.038
Or, if you want, you could factor out

00:04:01.038 --> 00:04:02.549
an e to the x.

00:04:02.549 --> 00:04:05.347
This is the same thing as

00:04:05.347 --> 00:04:09.514
e to the x times cosine
of x minus sine of x.

00:04:10.793 --> 00:04:14.184
Cosine of x minus sine of x.

00:04:14.184 --> 00:04:17.246
So hopefully this makes the product rule

00:04:17.246 --> 00:04:19.050
a little bit more tangible.

00:04:19.050 --> 00:04:20.421
And once you have this in your tool belt,

00:04:20.421 --> 00:04:23.228
there's a whole broader class of functions

00:04:23.228 --> 00:04:26.863
and expressions that we
can start to differentiate.


Worked example: Derivatives of sin(x) and cos(x)

WEBVTT
Kind: captions
Language: en

00:00:00.514 --> 00:00:01.714
- [Voiceover] What we
want to do is find the

00:00:01.714 --> 00:00:03.103
derivative of this G of X

00:00:03.103 --> 00:00:04.518
and at first it can look intimidating.

00:00:04.518 --> 00:00:05.584
We have a sine of X here.

00:00:05.584 --> 00:00:06.754
We have a cosine of X.

00:00:06.754 --> 00:00:09.102
We have this crazy expression here

00:00:09.102 --> 00:00:10.892
with a pi over cube root of X

00:00:10.892 --> 00:00:12.022
we're squaring the whole thing

00:00:12.022 --> 00:00:13.558
and at first it might seem intimidating.

00:00:13.558 --> 00:00:14.945
But as we'll see in this video, we can

00:00:14.945 --> 00:00:17.717
actually do this with the
tools already in our tool kit.

00:00:17.717 --> 00:00:20.007
Using our existing derivative properties

00:00:20.007 --> 00:00:21.953
using what we know about the power rule

00:00:21.953 --> 00:00:25.184
which tells us the
derivative with respect to X.

00:00:25.184 --> 00:00:26.267
Of X to the N

00:00:27.499 --> 00:00:29.166
is equal to N

00:00:29.166 --> 00:00:30.953
times X to the N minus one,

00:00:30.953 --> 00:00:32.558
we've see that multiple times.

00:00:32.558 --> 00:00:34.957
We also need to use the fact

00:00:34.957 --> 00:00:37.767
that the derivative of cosine of X

00:00:37.767 --> 00:00:40.035
is equal to negative the sine of X.

00:00:40.035 --> 00:00:41.449
And the other way around

00:00:41.449 --> 00:00:44.116
the derivative with respect to X

00:00:45.573 --> 00:00:47.035
of sine of X

00:00:47.035 --> 00:00:49.293
is equal to positive

00:00:49.293 --> 00:00:50.572
cosine of X.

00:00:50.572 --> 00:00:53.144
So using just that we can
actually evaluate this.

00:00:53.144 --> 00:00:55.291
Or evaluate G prime of X.

00:00:55.291 --> 00:00:57.880
So, pause the video and
see if you can do it.

00:00:57.880 --> 00:01:00.057
So probably the most
intimidating part of this

00:01:00.057 --> 00:01:01.643
because we know the derivative's a sine

00:01:01.643 --> 00:01:03.916
of X and cosine of X is
this expression here.

00:01:03.916 --> 00:01:05.844
And we can just rewrite this

00:01:05.844 --> 00:01:07.791
or simplify it a little
bit so it takes a form

00:01:07.791 --> 00:01:10.074
that you might be a little
bit more familiar with.

00:01:10.074 --> 00:01:12.225
So, so let me just do this

00:01:12.225 --> 00:01:13.713
on the side here.

00:01:13.713 --> 00:01:14.546
So, pi

00:01:15.684 --> 00:01:16.517
pi over

00:01:17.510 --> 00:01:19.010
the cube root of X

00:01:20.274 --> 00:01:21.506
squared.

00:01:21.506 --> 00:01:23.025
Well that's the same thing.

00:01:23.025 --> 00:01:25.275
This is equal to pi squared

00:01:26.174 --> 00:01:28.841
over the cube root of X squared.

00:01:30.342 --> 00:01:31.755
This is just exponent properties

00:01:31.755 --> 00:01:33.260
that we're dealing with.

00:01:33.260 --> 00:01:35.264
And so this is the same thing.

00:01:35.264 --> 00:01:37.470
We're gonna take X to the one third power

00:01:37.470 --> 00:01:39.949
and then raise that to the second power.

00:01:39.949 --> 00:01:42.780
So this is equal to pi squared

00:01:42.780 --> 00:01:44.166
over.

00:01:44.166 --> 00:01:44.999
Let me write it this way,

00:01:44.999 --> 00:01:45.981
I'm not gonna skip any steps

00:01:45.981 --> 00:01:48.387
because this is a good
review of exponent property.

00:01:48.387 --> 00:01:50.285
X to the one third squared.

00:01:50.285 --> 00:01:53.399
Which is the same thing as pi squared

00:01:53.399 --> 00:01:56.684
over X to the two thirds power.

00:01:56.684 --> 00:02:00.548
Which is the same thing as pi squared

00:02:00.548 --> 00:02:04.069
times X to the negative two thirds power.

00:02:04.069 --> 00:02:05.151
So when you write it like this

00:02:05.151 --> 00:02:06.681
it starts to get into a formula,

00:02:06.681 --> 00:02:08.448
you're like, oh, I can
see how the power rule

00:02:08.448 --> 00:02:09.366
could apply there.

00:02:09.366 --> 00:02:11.764
So this thing is just pi squared

00:02:11.764 --> 00:02:14.681
times X to the negative two thirds power.

00:02:14.681 --> 00:02:17.167
So actually let me delete this.

00:02:17.167 --> 00:02:18.330
So,

00:02:18.330 --> 00:02:19.163
this thing

00:02:20.746 --> 00:02:22.163
can be rewritten.

00:02:23.430 --> 00:02:24.831
This thing can be rewritten

00:02:24.831 --> 00:02:25.914
as pi squared

00:02:27.261 --> 00:02:29.178
times X to the negative

00:02:30.242 --> 00:02:32.992
to the negative two thirds power.

00:02:33.830 --> 00:02:35.022
So now let's take the derivative

00:02:35.022 --> 00:02:38.944
of each of these pieces
of this expression.

00:02:38.944 --> 00:02:40.505
So, we're gonna take

00:02:40.505 --> 00:02:43.585
we want to evaluate what
the G prime of X is.

00:02:43.585 --> 00:02:44.835
So G prime of X

00:02:45.766 --> 00:02:47.353
is going to be equal to.

00:02:47.353 --> 00:02:48.785
You could view it as a
derivative with respect

00:02:48.785 --> 00:02:51.570
to X of seven sine of X.

00:02:51.570 --> 00:02:52.562
So

00:02:52.562 --> 00:02:54.930
we can take do the derivative operator

00:02:54.930 --> 00:02:56.569
on both sides here just to make it clear

00:02:56.569 --> 00:02:57.977
what we're doing.

00:02:57.977 --> 00:02:59.490
So we're gonna apply it there.

00:02:59.490 --> 00:03:02.173
We're gonna apply it there.

00:03:02.173 --> 00:03:04.037
And we're going to apply it

00:03:04.037 --> 00:03:05.614
there.

00:03:05.614 --> 00:03:07.099
So this derivative

00:03:07.099 --> 00:03:09.347
this is the same thing as

00:03:09.347 --> 00:03:12.369
this is going to be seven
times the derivative

00:03:12.369 --> 00:03:13.878
of sine of X.

00:03:13.878 --> 00:03:16.215
So this is just gonna be seven times

00:03:16.215 --> 00:03:17.215
cosine of X.

00:03:19.227 --> 00:03:21.342
This one, over here,

00:03:21.342 --> 00:03:23.689
this is gonna be three,
or we're subtracting,

00:03:23.689 --> 00:03:25.402
so it's gonna be this subtract

00:03:25.402 --> 00:03:26.235
this minus.

00:03:26.235 --> 00:03:28.591
We can bring the constant out

00:03:28.591 --> 00:03:30.802
that we're multiplying the expression by.

00:03:30.802 --> 00:03:33.202
And the derivative of cosine of X

00:03:33.202 --> 00:03:35.255
so it's minus three times

00:03:35.255 --> 00:03:37.035
the derivative of cosine of X

00:03:37.035 --> 00:03:39.311
is negative sine of X.

00:03:39.311 --> 00:03:40.894
Negative sine of X.

00:03:42.131 --> 00:03:44.165
And then finally

00:03:44.165 --> 00:03:46.852
here in the yellow we
just apply the power rule.

00:03:46.852 --> 00:03:49.230
So, we have the negative two thirds,

00:03:49.230 --> 00:03:50.666
actually, let's not forget this minus sign

00:03:50.666 --> 00:03:52.108
I'm gonna write it out here.

00:03:52.108 --> 00:03:53.264
So you have the negative two thirds.

00:03:53.264 --> 00:03:55.566
You multiply the exponent
times the coefficient.

00:03:55.566 --> 00:03:56.876
It might look confusing, pi squared,

00:03:56.876 --> 00:03:58.054
but that's just a number.

00:03:58.054 --> 00:03:59.265
So it's gonna be

00:03:59.265 --> 00:04:00.136
negative

00:04:00.136 --> 00:04:02.410
and then you have negative two thirds

00:04:02.410 --> 00:04:03.827
times pi squared.

00:04:04.673 --> 00:04:06.090
Times pi squared.

00:04:07.023 --> 00:04:09.591
Times X to the negative two thirds

00:04:09.591 --> 00:04:11.017
minus one power.

00:04:11.017 --> 00:04:12.591
Negative two thirds

00:04:12.591 --> 00:04:13.922
minus one power.

00:04:13.922 --> 00:04:15.634
So what is this going to be?

00:04:15.634 --> 00:04:17.467
So we get G prime of X

00:04:18.792 --> 00:04:19.709
is equal to

00:04:20.762 --> 00:04:21.906
is equal to

00:04:21.906 --> 00:04:23.406
seven cosine of X.

00:04:24.827 --> 00:04:26.344
And let's see, we have a negative three

00:04:26.344 --> 00:04:27.829
times a negative sine of X.

00:04:27.829 --> 00:04:30.912
So that's a positive three sine of X.

00:04:32.550 --> 00:04:34.061
And then

00:04:34.061 --> 00:04:35.477
we have, we're subtracting

00:04:35.477 --> 00:04:36.771
but then this is going to be a negative,

00:04:36.771 --> 00:04:38.376
so that's going to be a positive.

00:04:38.376 --> 00:04:39.679
So we can say

00:04:39.679 --> 00:04:42.262
plus two pi squared over three.

00:04:43.461 --> 00:04:46.223
Two pi squared over three.

00:04:46.223 --> 00:04:47.744
That's that part there.

00:04:47.744 --> 00:04:48.577
Times X

00:04:49.957 --> 00:04:51.392
to the.

00:04:51.392 --> 00:04:52.747
So negative two thirds

00:04:52.747 --> 00:04:53.580
minus one,

00:04:54.723 --> 00:04:56.126
we could say negative one and two thirds,

00:04:56.126 --> 00:04:59.261
or we could say negative
five thirds by power.

00:04:59.261 --> 00:05:02.090
Negative five thirds power.

00:05:02.090 --> 00:05:02.923
And there you have it.

00:05:02.923 --> 00:05:04.006
We were able to tackle this thing that

00:05:04.006 --> 00:05:04.941
looked a little bit hairy

00:05:04.941 --> 00:05:07.158
but all we had to use was the power rule

00:05:07.158 --> 00:05:09.445
and what we knew to be the derivatives

00:05:09.445 --> 00:05:11.028
of sine and cosine.


Radical functions differentiation intro

WEBVTT
Kind: captions
Language: en

00:00:00.418 --> 00:00:03.298
- [Voiceover] Let's say that
we have a function, f of x,

00:00:03.298 --> 00:00:07.465
and it is equal to negative
four times the cube root

00:00:09.318 --> 00:00:11.808
times the cube root of x.

00:00:11.808 --> 00:00:14.141
And what we want to do
is we want to evaluate

00:00:14.141 --> 00:00:17.544
the derivative of our function
when x is equal to eight.

00:00:17.544 --> 00:00:19.755
So see if you can figure this out.

00:00:19.755 --> 00:00:22.181
All right, now this might
look foreign to you.

00:00:22.181 --> 00:00:23.552
You might say, well, I've
never taken a derivative

00:00:23.552 --> 00:00:25.738
of a cube root before, but as we'll see,

00:00:25.738 --> 00:00:28.650
we can actually just
apply the power rule here.

00:00:28.650 --> 00:00:31.715
Because this function can be rewritten as,

00:00:31.715 --> 00:00:35.538
so f of x can be rewritten
as negative four.

00:00:35.538 --> 00:00:37.453
The cube root of x is the same thing as

00:00:37.453 --> 00:00:39.933
x to the one-third power.

00:00:39.933 --> 00:00:41.404
And now it might be a little bit clearer

00:00:41.404 --> 00:00:43.095
that we can apply the power rule.

00:00:43.095 --> 00:00:45.592
We could take this one-third, multiply it

00:00:45.592 --> 00:00:48.158
by this co-efficient, negative four.

00:00:48.158 --> 00:00:51.846
So we have negative four times one-third,

00:00:51.846 --> 00:00:55.263
and you have times x to the one-third.

00:00:58.299 --> 00:01:00.201
We just decrement that exponent to,

00:01:00.201 --> 00:01:01.664
that's a different shade of blue,

00:01:01.664 --> 00:01:04.414
to the one-third minus one power.

00:01:06.364 --> 00:01:08.470
This is the derivative.

00:01:08.470 --> 00:01:11.470
So f prime of x is equal to that,

00:01:12.396 --> 00:01:13.950
and so now we just have to simplify.

00:01:13.950 --> 00:01:17.533
This is equal negative
four-thirds times x.

00:01:18.557 --> 00:01:21.909
One-third minus one is
negative two-thirds power.

00:01:21.909 --> 00:01:24.860
And so if we want to
evaluate f prime of eight,

00:01:24.860 --> 00:01:29.536
f prime of eight is equal
to negative four-thirds

00:01:29.536 --> 00:01:33.286
times eight to the
negative two-thirds power.

00:01:35.856 --> 00:01:39.180
Well that's the same thing
as negative four-thirds

00:01:39.180 --> 00:01:42.180
times eight, eight to the one-third,

00:01:45.020 --> 00:01:48.232
and then raise that to negative two power.

00:01:48.232 --> 00:01:49.715
I'm just using exponent properties here.

00:01:49.715 --> 00:01:51.598
If this looks completely
unfamiliar how I got

00:01:51.598 --> 00:01:52.753
from that to that, I encourage you

00:01:52.753 --> 00:01:55.427
to review exponent
properties on Khan Academy.

00:01:55.427 --> 00:01:58.747
Well eight to the
one-third, that is just two.

00:01:58.747 --> 00:02:02.881
So this is just two, and then
two to the negative two power.

00:02:02.881 --> 00:02:04.700
Remember, let me just
take some steps here.

00:02:04.700 --> 00:02:05.919
It's a good review.

00:02:05.919 --> 00:02:09.742
This is equal to negative four-thirds

00:02:09.742 --> 00:02:13.445
times two to the negative
two is the same thing

00:02:13.445 --> 00:02:15.695
as one over two to the two.

00:02:16.789 --> 00:02:18.763
These two things are equivalent.

00:02:18.763 --> 00:02:20.422
That and that.

00:02:20.422 --> 00:02:22.269
One over two to the two is the same thing

00:02:22.269 --> 00:02:23.658
as two to the negative two.

00:02:23.658 --> 00:02:26.186
So this is one over four,

00:02:26.186 --> 00:02:28.915
and so this is going to simplify to

00:02:28.915 --> 00:02:31.517
negative one over three.

00:02:31.517 --> 00:02:34.517
Is equal to negative one over three,

00:02:36.070 --> 00:02:37.403
and we are done.


Fractional powers differentiation

WEBVTT
Kind: captions
Language: en

00:00:00.301 --> 00:00:01.708
- [Voiceover] So we have h of x

00:00:01.708 --> 00:00:05.134
is equal to five x to the
1/4th power plus seven.

00:00:05.134 --> 00:00:07.627
And we wanna find what is h prime of 16,

00:00:07.627 --> 00:00:09.293
or what is the derivative of this function

00:00:09.293 --> 00:00:11.724
when x is equal to 16?

00:00:11.724 --> 00:00:12.971
And like always, pause this video

00:00:12.971 --> 00:00:15.378
and see if you can figure
it out on your own.

00:00:15.378 --> 00:00:16.268
All right, well let's just take

00:00:16.268 --> 00:00:17.998
the derivative of both sides of this.

00:00:17.998 --> 00:00:21.147
So on the left hand side,
I'm gonna have h prime of x,

00:00:21.147 --> 00:00:22.390
and on the right hand side,

00:00:22.390 --> 00:00:23.797
well the derivative of
the right hand side,

00:00:23.797 --> 00:00:25.843
I can just take the derivative
of five x to the 1/4th

00:00:25.843 --> 00:00:27.399
and add that to the derivative,

00:00:27.399 --> 00:00:29.732
with respect to x of seven.

00:00:29.732 --> 00:00:33.260
So the derivative of five
x to the 1/4th power,

00:00:33.260 --> 00:00:35.287
well, I can just apply
the power rule here.

00:00:35.287 --> 00:00:36.120
You might say, wait, wait wait,

00:00:36.120 --> 00:00:37.041
there's a fractional exponent,

00:00:37.041 --> 00:00:38.131
and I would just say, that's okay.

00:00:38.131 --> 00:00:40.636
The power rule is very powerful.

00:00:40.636 --> 00:00:43.872
So we can multiply the
1/4th times the coefficient.

00:00:43.872 --> 00:00:47.522
So you have five times 1/4th

00:00:47.522 --> 00:00:51.022
x to the 1/4th minus one power.

00:00:51.985 --> 00:00:54.769
That's the derivative of
five x to the 1/4th power.

00:00:54.769 --> 00:00:56.677
And then we have plus seven.

00:00:56.677 --> 00:00:59.030
Now what's the derivative of seven,

00:00:59.030 --> 00:01:00.527
with respect to x?

00:01:00.527 --> 00:01:02.535
Well seven doesn't
change with respect to x.

00:01:02.535 --> 00:01:03.504
The derivative of a constant,

00:01:03.504 --> 00:01:06.026
we've seen this multiple
times, is just zero.

00:01:06.026 --> 00:01:08.268
So it's just plus zero.

00:01:08.268 --> 00:01:10.285
And now we just have to simplify this.

00:01:10.285 --> 00:01:13.259
So this is gonna be h prime of x

00:01:13.259 --> 00:01:16.759
is equal to 5/4ths x to the,

00:01:18.526 --> 00:01:21.275
what's 1/4th minus one?

00:01:21.275 --> 00:01:23.164
Well that's negative 3/4ths.

00:01:23.164 --> 00:01:25.659
That's 1/4th minus 4/4ths,
or negative 3/4ths.

00:01:25.659 --> 00:01:29.826
So 5/4ths x to the
negative 3/4ths plus zero,

00:01:30.780 --> 00:01:32.241
so we don't have to write that.

00:01:32.241 --> 00:01:34.016
And now let's see if we can evaluate

00:01:34.016 --> 00:01:36.698
this when x is equal to 16.

00:01:36.698 --> 00:01:39.865
So, h prime of 16 is 5/4ths

00:01:43.317 --> 00:01:46.900
times 16 to the negative 3/4ths.

00:01:48.170 --> 00:01:50.144
Well that's the same thing as

00:01:50.144 --> 00:01:54.311
5/4ths times one over 16 to the 3/4ths,

00:01:58.200 --> 00:02:02.367
which is the same thing
as 5/4ths times one over,

00:02:03.696 --> 00:02:05.131
let's see, I could view this as

00:02:05.131 --> 00:02:09.214
16 to the 1/4th, and then cubing that.

00:02:10.701 --> 00:02:11.784
And so, what is this?

00:02:11.784 --> 00:02:14.919
16 to the 1/4th is two,

00:02:14.919 --> 00:02:16.533
and then you cube two.

00:02:16.533 --> 00:02:19.439
Two to the third power is
eight, so that's eight.

00:02:19.439 --> 00:02:22.144
So you have 5/4ths times 1/8th,

00:02:22.144 --> 00:02:23.787
which is going to be equal to

00:02:23.787 --> 00:02:25.974
five times one is five,

00:02:25.974 --> 00:02:28.557
and then four times eight is...

00:02:29.617 --> 00:02:32.000
four times eight is 32.

00:02:32.000 --> 00:02:33.333
And we are done.


Negative powers differentiation

WEBVTT
Kind: captions
Language: en

00:00:00.304 --> 00:00:02.219
- [Voiceover] So we have
the function g of x,

00:00:02.219 --> 00:00:05.864
which is equal to 2/x to
the third minus 1/x squared.

00:00:05.864 --> 00:00:07.817
And what I wanna do in this video,

00:00:07.817 --> 00:00:11.662
is I wanna find what g prime of x is

00:00:11.662 --> 00:00:15.376
and then I also wanna
evaluate that at x equal two,

00:00:15.376 --> 00:00:16.502
so I wanna figure that out.

00:00:16.502 --> 00:00:17.541
And I also wanna figure out what does

00:00:17.541 --> 00:00:19.737
that evaluate to when x is equal to two?

00:00:19.737 --> 00:00:21.132
So what is the slope
the of the tangent line

00:00:21.132 --> 00:00:24.947
to the graph of g, when x is equal to two?

00:00:24.947 --> 00:00:26.292
And like always, pause this video and

00:00:26.292 --> 00:00:28.251
see if you can work this out on your own

00:00:28.251 --> 00:00:30.668
before I work through it with you.

00:00:30.668 --> 00:00:31.850
And I'll give you some hints

00:00:31.850 --> 00:00:33.695
all you really need to do is

00:00:33.695 --> 00:00:36.450
apply the power rule, a little bit of

00:00:36.450 --> 00:00:38.084
basic exponent properties

00:00:38.084 --> 00:00:41.906
and some basic derivative properties

00:00:41.906 --> 00:00:43.729
to be able to do this.

00:00:43.729 --> 00:00:45.526
Alright, now let's just do this together

00:00:45.526 --> 00:00:46.495
and I'll just rewrite it.

00:00:46.495 --> 00:00:50.078
G of x is equal to this first term here,

00:00:51.378 --> 00:00:53.896
2/x to the third.

00:00:53.896 --> 00:00:56.362
Well, that could be rewritten as

00:00:56.362 --> 00:00:59.279
2 times x to the negative three.

00:01:00.624 --> 00:01:04.323
We know that 1/x to
the n is the same thing

00:01:04.323 --> 00:01:06.784
as x to the negative n.

00:01:06.784 --> 00:01:08.325
So I just rewrote it and this might

00:01:08.325 --> 00:01:10.255
be ringing a bell of how the power rule

00:01:10.255 --> 00:01:11.525
might be useful.

00:01:11.525 --> 00:01:13.451
And then we have minus-

00:01:13.451 --> 00:01:17.623
well, 1/x squared that is the same thing

00:01:17.623 --> 00:01:19.956
as x to the negative two.

00:01:21.030 --> 00:01:22.588
And so this, if we're gonna take

00:01:22.588 --> 00:01:24.486
the derivative of both sides of this,

00:01:24.486 --> 00:01:26.230
let's do that.

00:01:26.230 --> 00:01:28.647
Derivative with respect to x.

00:01:29.798 --> 00:01:32.527
Dx, we're gonna do that
on the left-hand side,

00:01:32.527 --> 00:01:36.374
we're also gonna do it
on the right-hand side.

00:01:36.374 --> 00:01:37.784
On the left-hand side, the derivative

00:01:37.784 --> 00:01:40.198
with respect to x of g of x,

00:01:40.198 --> 00:01:42.948
we can write that as g prime of x

00:01:44.680 --> 00:01:46.175
is going to be equal to,

00:01:46.175 --> 00:01:48.343
well, the derivative of this first

00:01:48.343 --> 00:01:50.356
that we have right here written in green,

00:01:50.356 --> 00:01:52.459
this is going to be, we're just gonna

00:01:52.459 --> 00:01:54.052
apply the power rule.

00:01:54.052 --> 00:01:57.255
We're going to take our exponent,

00:01:57.255 --> 00:02:00.070
multiply it by our coefficient out front,

00:02:00.070 --> 00:02:01.145
actually let me write that out,

00:02:01.145 --> 00:02:02.895
that's going to be...

00:02:04.059 --> 00:02:05.747
There's this equal sign.

00:02:05.747 --> 00:02:09.497
That is going to be two
times negative three,

00:02:11.803 --> 00:02:15.076
times x and now we're going
to decrement this exponent.

00:02:15.076 --> 00:02:16.277
You have to be very careful here

00:02:16.277 --> 00:02:17.427
because sometimes your brain might say,

00:02:17.427 --> 00:02:18.567
"Okay, one less than three is two,

00:02:18.567 --> 00:02:19.971
"so maybe this is x in the negative two."

00:02:19.971 --> 00:02:22.684
but remember, you're going down.

00:02:22.684 --> 00:02:25.233
So if you're at negative
three and you subtract one,

00:02:25.233 --> 00:02:26.478
we're gonna go with the negative three

00:02:26.478 --> 00:02:28.494
minus one power.

00:02:28.494 --> 00:02:30.167
We'll that's gonna take
us to negative four.

00:02:30.167 --> 00:02:33.577
So this is x to the negative four power.

00:02:33.577 --> 00:02:36.972
So two times negative three
x to the negative four,

00:02:36.972 --> 00:02:39.804
or we could have also written that as

00:02:39.804 --> 00:02:43.901
negative six x to the negative four power.

00:02:43.901 --> 00:02:45.401
And then, minus...

00:02:46.311 --> 00:02:47.884
Well, we're gonna do the same thing again

00:02:47.884 --> 00:02:49.049
right over here.

00:02:49.049 --> 00:02:52.466
We take this negative two, multiply it by

00:02:54.033 --> 00:02:55.683
the coefficient that's implicitly here,

00:02:55.683 --> 00:02:56.568
you could say there's a one there.

00:02:56.568 --> 00:02:58.142
So negative two times one.

00:02:58.142 --> 00:03:00.248
So you have the negative two there

00:03:00.248 --> 00:03:02.471
and then you have the x to the-

00:03:02.471 --> 00:03:04.755
well what's a negative two minus one?

00:03:04.755 --> 00:03:06.598
That's negative three.

00:03:06.598 --> 00:03:08.287
To the negative three power.

00:03:08.287 --> 00:03:10.952
And so we can rewrite
all of this business as,

00:03:10.952 --> 00:03:13.285
the derivative g prime of x,

00:03:14.250 --> 00:03:17.053
is equal to negative six,

00:03:17.053 --> 00:03:20.640
negative six x to the negative fourth.

00:03:20.640 --> 00:03:22.578
And now we're subtracting a negative.

00:03:22.578 --> 00:03:24.475
So we could just write this as,

00:03:24.475 --> 00:03:27.475
plus two x to the negative three.

00:03:29.114 --> 00:03:30.949
This negative cancels
out with that negative.

00:03:30.949 --> 00:03:33.366
Subtract a negative, same
thing as adding a positive.

00:03:33.366 --> 00:03:34.806
So we did the first part.

00:03:34.806 --> 00:03:38.438
We can express g prime
of x as a function of x.

00:03:38.438 --> 00:03:41.663
Now, let's just evaluate
what g prime of two is.

00:03:41.663 --> 00:03:45.080
So g prime of two is going to be equal to

00:03:46.300 --> 00:03:50.467
negative six times two to
the negative fourth power

00:03:51.732 --> 00:03:56.188
plus two times two to
the negative third power.

00:03:56.188 --> 00:03:57.926
Well, what's this going to be?

00:03:57.926 --> 00:04:01.810
This is equal to negative
6/2 to the fourth,

00:04:01.810 --> 00:04:04.560
plus 2/2 to the third,

00:04:05.491 --> 00:04:09.319
which is equal to negative six over-

00:04:09.319 --> 00:04:11.274
two to the fourth is 16,

00:04:11.274 --> 00:04:15.284
plus 2/2 to the third is eight.

00:04:15.284 --> 00:04:17.617
And so let's see, this is...

00:04:18.521 --> 00:04:20.662
Lets rewrite this all
with a common denominator.

00:04:20.662 --> 00:04:22.179
I could write this as 1/4,

00:04:22.179 --> 00:04:25.105
but then this one won't
work out as cleanly,

00:04:25.105 --> 00:04:26.874
I could write them both as eights.

00:04:26.874 --> 00:04:28.708
This is negative 3/8s.

00:04:28.708 --> 00:04:31.065
Negative 3/8s.

00:04:31.065 --> 00:04:33.929
So you have negative 3/8s plus two eights

00:04:33.929 --> 00:04:37.432
is equal to negative 1/8.

00:04:37.432 --> 00:04:40.976
So the slope of the tangent
line at x equals two,

00:04:40.976 --> 00:04:43.987
to the graph y equals g of x has a slope.

00:04:43.987 --> 00:04:47.154
That slope is negative 1/8.


Tangents of polynomials

WEBVTT
Kind: captions
Language: en

00:00:00.636 --> 00:00:02.173
- [Voiceover] What you see here in blue

00:00:02.173 --> 00:00:03.549
this is the graph of

00:00:03.549 --> 00:00:05.299
Y is equal to F of X.

00:00:07.140 --> 00:00:09.577
Where F of X is equal to

00:00:09.577 --> 00:00:11.434
X to the third

00:00:11.434 --> 00:00:12.790
minus six X squared

00:00:12.790 --> 00:00:14.819
plus X minus five.

00:00:14.819 --> 00:00:16.721
What I want to do in this video

00:00:16.721 --> 00:00:19.455
is think about what is the equation

00:00:19.455 --> 00:00:21.304
of the tangent line

00:00:21.304 --> 00:00:23.495
when X is equal to one?

00:00:23.495 --> 00:00:25.339
So we can visualize that.

00:00:25.339 --> 00:00:28.896
So, this is X equaling
one right over here.

00:00:28.896 --> 00:00:31.073
This is the value of the function.

00:00:31.073 --> 00:00:33.321
When X is equal to one.

00:00:33.321 --> 00:00:34.275
Right over there.

00:00:34.275 --> 00:00:35.758
And then the tangent line

00:00:35.758 --> 00:00:37.870
looks something like

00:00:37.870 --> 00:00:39.772
will look something like.

00:00:39.772 --> 00:00:41.635
I know I can do a better job than that.

00:00:41.635 --> 00:00:44.802
It's going to look something like that.

00:00:44.802 --> 00:00:48.076
And what we want to do
is find the equation

00:00:48.076 --> 00:00:50.825
the equation of that line.

00:00:50.825 --> 00:00:53.345
And if you are inspired
I encourage you to be,

00:00:53.345 --> 00:00:56.379
pause the video and try to work it out.

00:00:56.379 --> 00:00:58.039
Well the way that we can do this is

00:00:58.039 --> 00:01:00.793
if we find the derivative at X equals one

00:01:00.793 --> 00:01:03.882
the derivative is the
slope of the tangent line.

00:01:03.882 --> 00:01:05.860
And so we'll know the
slope of the tangent line.

00:01:05.860 --> 00:01:08.329
And we know that it contains that point

00:01:08.329 --> 00:01:10.290
and then we can use that
to find the equation

00:01:10.290 --> 00:01:11.640
of the tangent line.

00:01:11.640 --> 00:01:13.093
So let's actually just, let's just.

00:01:13.093 --> 00:01:14.539
So we want the equation
of the tangent line

00:01:14.539 --> 00:01:16.172
when X is equal to one.

00:01:16.172 --> 00:01:19.054
So let's just first of
all evaluate F of one.

00:01:19.054 --> 00:01:19.971
So F of one

00:01:20.806 --> 00:01:23.527
is equal to one to the third power

00:01:23.527 --> 00:01:24.805
which is one.

00:01:24.805 --> 00:01:26.358
Minus six times ones squared,

00:01:26.358 --> 00:01:28.500
so it's just minus six.

00:01:28.500 --> 00:01:29.514
And then

00:01:29.514 --> 00:01:30.858
plus one

00:01:30.858 --> 00:01:32.186
plus one

00:01:32.186 --> 00:01:33.103
minus five.

00:01:34.017 --> 00:01:36.604
So, this is equal to what?

00:01:36.604 --> 00:01:37.687
Two minus 11?

00:01:39.266 --> 00:01:41.017
Which is equal to

00:01:41.017 --> 00:01:42.506
negative nine.

00:01:42.506 --> 00:01:43.737
And that looks about right.

00:01:43.737 --> 00:01:45.675
That looks like about negative
nine right over there.

00:01:45.675 --> 00:01:48.783
The scales are different
on the Y and the X axis.

00:01:48.783 --> 00:01:49.969
So that is F of one.

00:01:49.969 --> 00:01:51.866
It is negative nine.

00:01:51.866 --> 00:01:52.699
Did I do that right?

00:01:52.699 --> 00:01:54.045
This is negative five.

00:01:54.045 --> 00:01:55.324
Yep, negative nine.

00:01:55.324 --> 00:01:57.923
And now let's evaluate
what the derivative is

00:01:57.923 --> 00:01:59.139
at one.

00:01:59.139 --> 00:02:00.018
So,

00:02:00.018 --> 00:02:01.768
what is F prime of X?

00:02:03.295 --> 00:02:04.378
F prime of X.

00:02:06.693 --> 00:02:08.862
Well here it's just a polynomial.

00:02:08.862 --> 00:02:10.798
You take the derivative of X to the third

00:02:10.798 --> 00:02:13.547
while we apply the power rule.

00:02:13.547 --> 00:02:16.027
We bring the three out front.

00:02:16.027 --> 00:02:18.063
So you get three X

00:02:18.063 --> 00:02:18.896
to the.

00:02:18.896 --> 00:02:21.208
And then we go one less than three

00:02:21.208 --> 00:02:23.272
to get the second power.

00:02:23.272 --> 00:02:24.355
Then you have

00:02:25.605 --> 00:02:27.503
minus six X squared.

00:02:27.503 --> 00:02:29.784
So you bring the two times the six

00:02:29.784 --> 00:02:31.160
to get 12.

00:02:31.160 --> 00:02:32.243
So minus 12 X

00:02:34.552 --> 00:02:35.887
to the

00:02:35.887 --> 00:02:37.920
well two minus one is one power

00:02:37.920 --> 00:02:40.141
so that's the same thing as 12 X.

00:02:40.141 --> 00:02:43.293
And then plus the derivative of X

00:02:43.293 --> 00:02:45.116
is just one.

00:02:45.116 --> 00:02:46.596
That's just going to be one.

00:02:46.596 --> 00:02:48.734
And if you view this
as X to the first power

00:02:48.734 --> 00:02:50.662
we're just bringing the one out front

00:02:50.662 --> 00:02:52.664
and decrementing the one.

00:02:52.664 --> 00:02:55.028
So it's one times X to the zero power

00:02:55.028 --> 00:02:56.428
which is just one.

00:02:56.428 --> 00:02:58.881
And then the derivative of a constant here

00:02:58.881 --> 00:03:00.780
is just going to be zero.

00:03:00.780 --> 00:03:02.468
So this is our derivative of F

00:03:02.468 --> 00:03:04.364
and if we want to evaluate it at one

00:03:04.364 --> 00:03:05.531
F prime of one

00:03:06.581 --> 00:03:08.684
is going to be three times one squared

00:03:08.684 --> 00:03:09.965
which is just three

00:03:09.965 --> 00:03:11.397
minus 12 times one

00:03:11.397 --> 00:03:13.187
which is just minus 12.

00:03:13.187 --> 00:03:15.807
And then we have plus one.

00:03:15.807 --> 00:03:17.050
So this is

00:03:17.050 --> 00:03:19.633
three minus 12 is negative nine

00:03:20.628 --> 00:03:23.802
plus one is equal to negative eight.

00:03:23.802 --> 00:03:25.646
So we know the slope right over here

00:03:25.646 --> 00:03:27.084
is the slope of negative eight.

00:03:27.084 --> 00:03:28.745
We know a point on that line

00:03:28.745 --> 00:03:30.829
it contains the point one negative nine

00:03:30.829 --> 00:03:33.109
so we can use that information to find the

00:03:33.109 --> 00:03:34.162
equation of the line.

00:03:34.162 --> 00:03:35.746
The line, just to remind ourselves,

00:03:35.746 --> 00:03:36.949
has the four.

00:03:36.949 --> 00:03:39.032
Y is equal to M X plus B.

00:03:40.513 --> 00:03:42.195
Where M is the slope.

00:03:42.195 --> 00:03:44.032
So we know that Y is going to be

00:03:44.032 --> 00:03:46.165
equal to negative eight X

00:03:46.165 --> 00:03:46.998
plus B.

00:03:48.543 --> 00:03:49.714
And now we can substitute

00:03:49.714 --> 00:03:51.253
the X and Y value that we know

00:03:51.253 --> 00:03:53.206
sits on that line to solve for B.

00:03:53.206 --> 00:03:56.238
So we know that Y is
equal to negative nine.

00:03:56.238 --> 00:03:57.564
Let me just write this here.

00:03:57.564 --> 00:03:59.431
Y is equal to negative nine

00:03:59.431 --> 00:04:00.931
when X is equal to

00:04:02.143 --> 00:04:03.895
when X is equal to one.

00:04:03.895 --> 00:04:04.978
And so we get

00:04:06.208 --> 00:04:07.503
we get

00:04:07.503 --> 00:04:08.578
negative nine

00:04:08.578 --> 00:04:10.828
is equal to negative eight times one.

00:04:10.828 --> 00:04:12.325
So negative eight

00:04:12.325 --> 00:04:13.158
plus B.

00:04:14.024 --> 00:04:15.543
Well, let's see.

00:04:15.543 --> 00:04:16.543
We could add

00:04:17.450 --> 00:04:19.381
we could eight to both sides

00:04:19.381 --> 00:04:21.151
and we get negative one

00:04:21.151 --> 00:04:22.982
is equal to B.

00:04:22.982 --> 00:04:23.815
So we're done.

00:04:23.815 --> 00:04:25.408
The equation of the line

00:04:25.408 --> 00:04:26.612
the equation of this line

00:04:26.612 --> 00:04:28.831
that we have in magenta right over there

00:04:28.831 --> 00:04:29.664
that is

00:04:30.753 --> 00:04:32.000
that is

00:04:32.000 --> 00:04:33.917
Y is equal to the slope

00:04:35.288 --> 00:04:37.330
is negative eight X.

00:04:37.330 --> 00:04:38.931
And then the Y-intercept

00:04:38.931 --> 00:04:39.764
minus one.


Differentiating polynomials example

WEBVTT
Kind: captions
Language: en

00:00:00.647 --> 00:00:02.658
- [Voiceover] So I have
the function F of X here

00:00:02.658 --> 00:00:05.872
and we're defining it using
a polynomial expression.

00:00:05.872 --> 00:00:07.479
And what I would like to do here is

00:00:07.479 --> 00:00:09.493
take the derivative of our function

00:00:09.493 --> 00:00:10.910
which is essentially gonna make us take

00:00:10.910 --> 00:00:12.964
the derivative of this
polynomial expression

00:00:12.964 --> 00:00:14.323
and we're gonna take the derivative

00:00:14.323 --> 00:00:16.017
with respect to X.

00:00:16.017 --> 00:00:17.429
So the first thing I'm gonna do is

00:00:17.429 --> 00:00:19.910
let's take the derivative of both sides.

00:00:19.910 --> 00:00:22.846
So we can say the
derivative with respect to X

00:00:22.846 --> 00:00:23.679
of F of X

00:00:24.831 --> 00:00:25.664
of F of X

00:00:27.191 --> 00:00:29.025
is equal to the derivative

00:00:29.025 --> 00:00:30.577
with respect to X

00:00:30.577 --> 00:00:32.908
the derivative with respect to X

00:00:32.908 --> 00:00:34.325
of X to the fifth

00:00:35.894 --> 00:00:37.576
X to the fifth

00:00:37.576 --> 00:00:38.409
plus two

00:00:39.561 --> 00:00:41.478
plus two X to the third

00:00:43.678 --> 00:00:45.011
minus X squared.

00:00:46.974 --> 00:00:49.511
And so the notation, just
to get familiar with it

00:00:49.511 --> 00:00:51.323
you could do this as
the derivative operator.

00:00:51.323 --> 00:00:52.698
This is, I want to take the derivative

00:00:52.698 --> 00:00:54.928
of whatever's inside of the parentheses

00:00:54.928 --> 00:00:56.956
with respect to X.

00:00:56.956 --> 00:00:59.342
So the derivative of F with respect to X

00:00:59.342 --> 00:01:00.614
we could use the notation

00:01:00.614 --> 00:01:02.864
but that is just F prime of

00:01:03.961 --> 00:01:05.562
F prime of X.

00:01:05.562 --> 00:01:08.719
And that is going to be equal to.

00:01:08.719 --> 00:01:11.318
Now here we can use our
derivative properties.

00:01:11.318 --> 00:01:13.123
The derivative of the sum or difference

00:01:13.123 --> 00:01:14.224
of a bunch of things.

00:01:14.224 --> 00:01:16.911
The derivative of is equal to the sum

00:01:16.911 --> 00:01:19.246
of the difference of the
derivative of each of them.

00:01:19.246 --> 00:01:22.290
So this is equal to the derivative

00:01:22.290 --> 00:01:24.960
let me just, with the
derivative with respect to X

00:01:24.960 --> 00:01:26.593
of each of these three things.

00:01:26.593 --> 00:01:28.991
So the derivative with respect to X.

00:01:28.991 --> 00:01:31.135
Let me just write it out like this.

00:01:31.135 --> 00:01:32.821
Of that first term

00:01:32.821 --> 00:01:36.228
plus the derivative with respect to X

00:01:36.228 --> 00:01:38.012
of that second term

00:01:38.012 --> 00:01:41.716
minus the derivative with respect to X

00:01:41.716 --> 00:01:43.785
of that third term.

00:01:43.785 --> 00:01:45.471
Of that third term.

00:01:45.471 --> 00:01:47.352
And I'll color code it here.

00:01:47.352 --> 00:01:49.135
So here I had an X of the fifth

00:01:49.135 --> 00:01:50.364
so I'll put the X to fifth

00:01:50.364 --> 00:01:51.921
X of the fifth there.

00:01:51.921 --> 00:01:54.588
Here I had a two X to the third.

00:01:57.056 --> 00:02:00.386
So I'll put the two X to the third there.

00:02:00.386 --> 00:02:03.013
And here I have a X squared

00:02:03.013 --> 00:02:04.147
I'm subtracting an X squared

00:02:04.147 --> 00:02:06.792
so I'm subtracting the
derivative with respect to X

00:02:06.792 --> 00:02:08.068
of X squared.

00:02:08.068 --> 00:02:10.162
So notice all that's happening here is

00:02:10.162 --> 00:02:12.298
I'm taking the derivative individually of

00:02:12.298 --> 00:02:13.792
each of these terms and then I'm adding

00:02:13.792 --> 00:02:15.396
or subtracting in the
same way that the terms

00:02:15.396 --> 00:02:18.297
were added or subtracted.

00:02:18.297 --> 00:02:20.625
And so what is this going to be equal to?

00:02:20.625 --> 00:02:22.658
Well, this is going to be equal to

00:02:22.658 --> 00:02:23.787
for X of the fifth

00:02:23.787 --> 00:02:26.015
we can just use the power rule.

00:02:26.015 --> 00:02:28.265
We can bring the five out front

00:02:28.265 --> 00:02:30.743
and decrement the exponent by one

00:02:30.743 --> 00:02:32.410
so it becomes five X

00:02:33.825 --> 00:02:36.678
we can say to the five minus one power.

00:02:36.678 --> 00:02:39.579
Which of course, is just four.

00:02:39.579 --> 00:02:41.958
And then for this second one

00:02:41.958 --> 00:02:44.007
we could do it in a few steps.

00:02:44.007 --> 00:02:45.991
Actually, let me just write it out here.

00:02:45.991 --> 00:02:47.666
So I could write

00:02:47.666 --> 00:02:49.459
I could write the derivative

00:02:49.459 --> 00:02:51.178
with respect to X

00:02:51.178 --> 00:02:54.196
of two X to the third power

00:02:54.196 --> 00:02:56.047
is the same thing

00:02:56.047 --> 00:02:57.595
it's equal to

00:02:57.595 --> 00:02:59.498
the same, we could bring the constant out.

00:02:59.498 --> 00:03:01.824
The derivative with,
two times the derivative

00:03:01.824 --> 00:03:02.946
with respect to X

00:03:02.946 --> 00:03:04.893
of X to the third power.

00:03:04.893 --> 00:03:06.093
This is one of our

00:03:06.093 --> 00:03:07.660
this is one of our derivative properties.

00:03:07.660 --> 00:03:09.062
The derivative of a constant

00:03:09.062 --> 00:03:10.313
times some expression

00:03:10.313 --> 00:03:12.053
is the same thing as a constant

00:03:12.053 --> 00:03:14.561
times the derivative of that expression.

00:03:14.561 --> 00:03:17.053
And what will the
derivative with respect to X

00:03:17.053 --> 00:03:18.798
to the third be?

00:03:18.798 --> 00:03:20.946
Well, we would bring the three out front

00:03:20.946 --> 00:03:22.736
and decrement the exponent.

00:03:22.736 --> 00:03:24.997
And so this would be equal to

00:03:24.997 --> 00:03:26.282
this two

00:03:26.282 --> 00:03:27.532
times the three

00:03:28.961 --> 00:03:32.044
times X to the three minus one power.

00:03:33.644 --> 00:03:37.038
Which is, of course, the second power.

00:03:37.038 --> 00:03:40.795
So this would give us six X squared.

00:03:40.795 --> 00:03:42.453
So, another way that
you could have done it

00:03:42.453 --> 00:03:43.866
I could just write

00:03:43.866 --> 00:03:46.048
I could just write a six X squared here.

00:03:46.048 --> 00:03:48.538
So I could just, so this is going to be

00:03:48.538 --> 00:03:49.705
six X squared.

00:03:51.053 --> 00:03:53.098
And, instead of going through all of this

00:03:53.098 --> 00:03:54.715
you'll learn as you do more of these,

00:03:54.715 --> 00:03:56.717
that you could have done this pretty much

00:03:56.717 --> 00:03:58.148
in your head.

00:03:58.148 --> 00:04:00.749
Say look, I have the three
out here as an exponent.

00:04:00.749 --> 00:04:03.483
Let me multiply the three
times this coefficient

00:04:03.483 --> 00:04:05.324
because that's what we
ended up doing anyway.

00:04:05.324 --> 00:04:08.691
Three times the coefficient is six X

00:04:08.691 --> 00:04:12.092
and then three minus one is two.

00:04:12.092 --> 00:04:13.650
So you didn't necessarily have to do this

00:04:13.650 --> 00:04:15.722
but it's nice to see
that this comes out of

00:04:15.722 --> 00:04:17.298
the derivative properties
that we talk about

00:04:17.298 --> 00:04:18.688
in other videos.

00:04:18.688 --> 00:04:20.644
And then finally,

00:04:20.644 --> 00:04:22.185
we have minus.

00:04:22.185 --> 00:04:23.352
And we use the

00:04:24.190 --> 00:04:26.370
power rule right over here.

00:04:26.370 --> 00:04:27.909
So, bring the two out front

00:04:27.909 --> 00:04:29.221
and decrement the exponent.

00:04:29.221 --> 00:04:30.936
So it's gonna be two.

00:04:30.936 --> 00:04:32.269
It's going to be

00:04:33.353 --> 00:04:34.270
two times X

00:04:35.857 --> 00:04:37.463
to the two minus one power

00:04:37.463 --> 00:04:38.963
which is just one.

00:04:39.897 --> 00:04:43.292
Which, we could just write as two X.

00:04:43.292 --> 00:04:44.840
So just like that we have been able

00:04:44.840 --> 00:04:47.956
to figure out the derivative of F.

00:04:47.956 --> 00:04:50.009
And you might say, well
what this thing now?

00:04:50.009 --> 00:04:51.870
Well now we have an
expression that tells us

00:04:51.870 --> 00:04:53.810
the slope of the tangent line.

00:04:53.810 --> 00:04:55.204
Or you could view it as the instantaneous

00:04:55.204 --> 00:04:57.322
rate of change with respect to X

00:04:57.322 --> 00:04:58.655
for any X value.

00:04:59.954 --> 00:05:02.426
So, if I were to say

00:05:02.426 --> 00:05:04.268
if I were to now to say

00:05:04.268 --> 00:05:07.101
F prime, let's say F prime of two.

00:05:08.765 --> 00:05:10.788
This would tell me what is

00:05:10.788 --> 00:05:13.330
the slope of the tangent
line of our function

00:05:13.330 --> 00:05:14.940
when X is equal to two.

00:05:14.940 --> 00:05:17.339
And I do that by using this expression.

00:05:17.339 --> 00:05:19.365
So this is gonna be

00:05:19.365 --> 00:05:21.698
five times two to the fourth

00:05:22.829 --> 00:05:23.662
plus plus

00:05:25.907 --> 00:05:27.740
six times two squared.

00:05:28.700 --> 00:05:30.533
Six times two squared.

00:05:31.521 --> 00:05:32.438
Minus minus

00:05:33.971 --> 00:05:35.690
two times two.

00:05:35.690 --> 00:05:37.357
Minus two times two.

00:05:38.511 --> 00:05:41.290
And this is going to be equal to

00:05:41.290 --> 00:05:44.060
let's see, two to the fourth power is 16,

00:05:44.060 --> 00:05:45.727
16 times five is 80.

00:05:47.674 --> 00:05:49.591
And that's 80, and then

00:05:50.540 --> 00:05:52.236
this is six times four,

00:05:52.236 --> 00:05:53.236
which is 24.

00:05:54.402 --> 00:05:58.290
And then we are going to subtract four.

00:05:58.290 --> 00:06:00.623
So this is 80 plus 24 is 104

00:06:01.921 --> 00:06:04.171
minus four is equal to 100.

00:06:05.691 --> 00:06:07.858
So, when X is equal to two

00:06:09.038 --> 00:06:11.602
this curve is really steep.

00:06:11.602 --> 00:06:13.583
The slope is 100.

00:06:13.583 --> 00:06:15.154
If you look, if you were to graph

00:06:15.154 --> 00:06:18.024
the tangent line when X is equal to two

00:06:18.024 --> 00:06:22.270
for every positive
movement in the X-direction

00:06:22.270 --> 00:06:24.966
by one you're gonna move
up in the Y-direction

00:06:24.966 --> 00:06:25.875
by 100.

00:06:25.875 --> 00:06:27.674
So it's really steep
there and it makes sense.

00:06:27.674 --> 00:06:29.251
This is a pretty high degree.

00:06:29.251 --> 00:06:31.236
X to the fifth power and
then we're adding that

00:06:31.236 --> 00:06:32.422
to another high degree

00:06:32.422 --> 00:06:33.558
X to the third power.

00:06:33.558 --> 00:06:35.523
And then we're subtracting a lower degree.

00:06:35.523 --> 00:06:37.870
So that's what you would expect.


Basic derivative rules: find the error

WEBVTT
Kind: captions
Language: en

00:00:00.230 --> 00:00:02.093
- [Voiceover] So we have two examples here

00:00:02.093 --> 00:00:04.796
of someone trying to find the
derivative of an expression.

00:00:04.796 --> 00:00:06.539
On the left-hand side, it says

00:00:06.539 --> 00:00:08.315
"Avery tried to find the derivative,

00:00:08.315 --> 00:00:11.934
"of seven minus five x using
basic differentiation rules.

00:00:11.934 --> 00:00:13.589
"Here is her work,"

00:00:13.589 --> 00:00:14.539
and on the right-hand side it says

00:00:14.539 --> 00:00:15.782
"Hannah tried to find the derivative,

00:00:15.782 --> 00:00:17.628
"of negative three plus eight x,

00:00:17.628 --> 00:00:20.096
"using basic differentiation rules,

00:00:20.096 --> 00:00:21.529
"here is her work."

00:00:21.529 --> 00:00:24.138
And these are two different examples of

00:00:24.138 --> 00:00:26.818
differentiation rules
exercise on Khan academy,

00:00:26.818 --> 00:00:28.730
and I thought I would
just do them side by side,

00:00:28.730 --> 00:00:31.653
because we can kind of think
about what each of these

00:00:31.653 --> 00:00:34.380
people are doing correct or incorrect.

00:00:34.380 --> 00:00:36.307
So these are similar expressions,

00:00:36.307 --> 00:00:38.989
we have a constant and then
we have a first degree term,

00:00:38.989 --> 00:00:41.042
a constant and then first degree term.

00:00:41.042 --> 00:00:42.466
So they're gonna take the derivative,

00:00:42.466 --> 00:00:44.883
so let's see, step one for Avery.

00:00:44.883 --> 00:00:48.324
She took, she's separately
taking the derivative of seven,

00:00:48.324 --> 00:00:52.241
and separately taking
the derivative of five x.

00:00:53.770 --> 00:00:58.599
So this my spider senses
already going off here,

00:00:58.599 --> 00:01:02.766
because what happened to this
negative right over here?

00:01:03.685 --> 00:01:07.503
So it would've sense for
her to do the derivative

00:01:07.503 --> 00:01:12.321
of seven, and she could've
said minus the derivative,

00:01:12.321 --> 00:01:16.732
of five x, that's one possibility
that she could've done.

00:01:16.732 --> 00:01:18.678
The derivative of a difference is equal to

00:01:18.678 --> 00:01:20.402
the difference of the derivatives,

00:01:20.402 --> 00:01:21.869
we've seen that property.

00:01:21.869 --> 00:01:23.274
Or, she could've said,

00:01:23.274 --> 00:01:25.824
the derivative, she could've
said this was equal to

00:01:25.824 --> 00:01:27.819
the derivative of seven,

00:01:27.819 --> 00:01:29.402
plus the derivative

00:01:30.284 --> 00:01:33.717
with respect to x of negative five x.

00:01:33.717 --> 00:01:36.292
These two things would've
been equivalent to this one.

00:01:36.292 --> 00:01:39.528
But for this one, she somehow forgot to,

00:01:39.528 --> 00:01:40.987
include the negative.

00:01:40.987 --> 00:01:45.398
So I think she had a
problem right at step one.

00:01:45.398 --> 00:01:48.877
Now, if you just follow
her logic after step one,

00:01:48.877 --> 00:01:51.058
let's see if she makes any more mistakes.

00:01:51.058 --> 00:01:54.329
So, she takes the
derivative of a constant,

00:01:54.329 --> 00:01:57.060
so constant isn't going to
change with respect to x.

00:01:57.060 --> 00:01:59.749
So that makes sense, that
that derivative is zero.

00:01:59.749 --> 00:02:02.320
And so we still have the
derivative of five x,

00:02:02.320 --> 00:02:05.299
and remember, it should've
been negative five x,

00:02:05.299 --> 00:02:07.702
or minus the derivative of five x.

00:02:07.702 --> 00:02:09.022
And let's see what she does here.

00:02:09.022 --> 00:02:10.719
So that zero disappears,

00:02:10.719 --> 00:02:12.989
and now she takes the constant,

00:02:12.989 --> 00:02:14.311
she takes the constant out,

00:02:14.311 --> 00:02:15.144
and that's true,

00:02:15.144 --> 00:02:17.019
the derivative of a
constant times something,

00:02:17.019 --> 00:02:18.354
is equal to the constant

00:02:18.354 --> 00:02:20.681
times the derivative of that something.

00:02:20.681 --> 00:02:23.995
And then, she finds the
derivative with respect to x

00:02:23.995 --> 00:02:25.724
of x is one, and that's true,

00:02:25.724 --> 00:02:28.405
if the slope, if you had
the graph of y equals x,

00:02:28.405 --> 00:02:29.597
the slope there is one,

00:02:29.597 --> 00:02:30.561
or what's the rate of change

00:02:30.561 --> 00:02:32.765
at which x changes with respect to x?

00:02:32.765 --> 00:02:35.017
Well, that's gonna be one for one.

00:02:35.017 --> 00:02:36.447
So that the slope here is one,

00:02:36.447 --> 00:02:38.267
so this is gonna be five times one.

00:02:38.267 --> 00:02:40.164
Which is equal to five,

00:02:40.164 --> 00:02:40.997
and at the end they just say,

00:02:40.997 --> 00:02:43.227
at what step did Avery make a mistake?

00:02:43.227 --> 00:02:46.171
So she clearly made a mistake at step one,

00:02:46.171 --> 00:02:49.148
this right of here,
should've been a negative,

00:02:49.148 --> 00:02:51.642
that's a negative, then that
would've been a negative.

00:02:51.642 --> 00:02:53.236
And this would've been a negative.

00:02:53.236 --> 00:02:54.581
And that would've been a negative.

00:02:54.581 --> 00:02:56.773
And then her final
answer should have been,

00:02:56.773 --> 00:02:58.904
should have been a negative five.

00:02:58.904 --> 00:03:00.500
Now let's go back to Hannah.

00:03:00.500 --> 00:03:03.684
To see if she made any mistakes and where.

00:03:03.684 --> 00:03:05.933
So she's differentiating
a similar expression,

00:03:05.933 --> 00:03:08.502
so first she takes the
derivative of the constant,

00:03:08.502 --> 00:03:11.312
plus the derivative of
the first degree term.

00:03:11.312 --> 00:03:14.479
Derivative of constant
is zero, that looks good.

00:03:14.479 --> 00:03:15.408
So you get the zero,

00:03:15.408 --> 00:03:18.775
and then you have the derivative
of the first degree term.

00:03:18.775 --> 00:03:21.072
That's what she's trying to figure out.

00:03:21.072 --> 00:03:22.641
And then, let's see,

00:03:22.641 --> 00:03:24.427
she's taking...

00:03:24.427 --> 00:03:26.675
Let's see, so this seems off.

00:03:26.675 --> 00:03:29.522
She is assuming that the
derivative of a product

00:03:29.522 --> 00:03:32.535
is equal to the product
of the derivatives.

00:03:32.535 --> 00:03:34.829
That is not the case.

00:03:34.829 --> 00:03:37.783
And especially, and if
you have a constant here,

00:03:37.783 --> 00:03:39.788
there's actually a much simpler
way of thinking about it.

00:03:39.788 --> 00:03:41.804
Frankly the way that
Avery thought about it,

00:03:41.804 --> 00:03:43.594
Avery had made a mistake at step one,

00:03:43.594 --> 00:03:45.137
but this is actually going to be equal

00:03:45.137 --> 00:03:47.868
to the derivative of a
constant times an expression,

00:03:47.868 --> 00:03:50.367
is equal to the same thing as the constant

00:03:50.367 --> 00:03:52.367
times the derivative of,

00:03:53.702 --> 00:03:55.910
of the expression.

00:03:55.910 --> 00:03:57.755
So this would've been
the correct way to go,

00:03:57.755 --> 00:04:00.611
and the derivative of x with respect to x,

00:04:00.611 --> 00:04:02.357
well that's just going to be one.

00:04:02.357 --> 00:04:05.000
So this should've all simplified to eight.

00:04:05.000 --> 00:04:07.793
What she did is, she is assuming,

00:04:07.793 --> 00:04:09.174
she tried to take the derivative of eight

00:04:09.174 --> 00:04:10.750
and multiply that times
the derivative of x,

00:04:10.750 --> 00:04:12.310
that is not the way it works.

00:04:12.310 --> 00:04:13.792
In the future you will learn something

00:04:13.792 --> 00:04:15.305
called the product rule,

00:04:15.305 --> 00:04:16.810
but you won't even have
to apply that here,

00:04:16.810 --> 00:04:18.187
because one of these,

00:04:18.187 --> 00:04:19.908
one of these components
I guess you could say,

00:04:19.908 --> 00:04:20.999
is a constant.

00:04:20.999 --> 00:04:22.961
So this is the wrong step.

00:04:22.961 --> 00:04:25.199
This is where Hannah makes a mistake.

00:04:25.199 --> 00:04:26.032
And you could see,

00:04:26.032 --> 00:04:28.014
instead of getting a
final answer of eight,

00:04:28.014 --> 00:04:29.960
she is getting a final answer of,

00:04:29.960 --> 00:04:32.727
she assumes well the
derivative of eight is zero,

00:04:32.727 --> 00:04:34.165
times the derivative of x is one,

00:04:34.165 --> 00:04:35.193
zero times one.

00:04:35.193 --> 00:04:38.200
And she gets zero, which
is not the right answer.

00:04:38.200 --> 00:04:40.070
So she makes a mistake at step three,

00:04:40.070 --> 00:04:43.237
and Avery made a mistake at step one.


Basic derivative rules (Part 2)

WEBVTT
Kind: captions
Language: en

00:00:00.625 --> 00:00:02.910
- [Voiceover] In the last
video we introduced you to

00:00:02.910 --> 00:00:04.813
the derivative property right over here

00:00:04.813 --> 00:00:06.797
that if my function is
equal to some constant

00:00:06.797 --> 00:00:09.397
that the derivative is going to be zero

00:00:09.397 --> 00:00:11.963
at any X, and we made a graphical argument

00:00:11.963 --> 00:00:14.056
and we also used the
definition of the limits

00:00:14.056 --> 00:00:15.742
to feel good about that.

00:00:15.742 --> 00:00:18.173
Now let's give a few
more of these properties

00:00:18.173 --> 00:00:19.765
and these are core properties as you

00:00:19.765 --> 00:00:22.025
throughout the rest of your calculus life

00:00:22.025 --> 00:00:24.449
your career, you will be
using some combination

00:00:24.449 --> 00:00:26.699
of these properties to find derivatives.

00:00:26.699 --> 00:00:28.356
So it's good to, one, know about them,

00:00:28.356 --> 00:00:30.710
and to feel good that
they're actually true.

00:00:30.710 --> 00:00:33.293
The second one is if my function F of X

00:00:33.293 --> 00:00:35.327
is equal to some constant

00:00:35.327 --> 00:00:37.710
times another function G of X

00:00:37.710 --> 00:00:40.331
well, then the derivative

00:00:40.331 --> 00:00:42.746
of F of X is going to be equal

00:00:42.746 --> 00:00:44.510
to that same constant.

00:00:44.510 --> 00:00:47.245
Let me do it, that same constant

00:00:47.245 --> 00:00:50.029
times the derivative of the G of X.

00:00:50.029 --> 00:00:52.385
And once again, maybe
we can make a actually

00:00:52.385 --> 00:00:55.473
we could make a graphical
argument for why that is true.

00:00:55.473 --> 00:00:58.390
This is going to multiply the slope

00:00:59.766 --> 00:01:01.206
is one way to think about it.

00:01:01.206 --> 00:01:02.809
But it's easier to make
an algebraic argument

00:01:02.809 --> 00:01:05.210
just using, frankly we can
use either one of these

00:01:05.210 --> 00:01:07.150
definitions of the derivative.

00:01:07.150 --> 00:01:08.456
I'll use the one on the right

00:01:08.456 --> 00:01:10.049
because it feels more general

00:01:10.049 --> 00:01:12.036
although you can say,
well this is true for A,

00:01:12.036 --> 00:01:13.460
but X could be any A.

00:01:13.460 --> 00:01:15.200
But I'll just use the one on the right.

00:01:15.200 --> 00:01:18.327
So, if we want to find F prime of X,

00:01:18.327 --> 00:01:21.440
F prime of X using this definition,

00:01:21.440 --> 00:01:22.857
we know, we know.

00:01:26.437 --> 00:01:29.069
F prime of X is going
to be equal to the limit

00:01:29.069 --> 00:01:32.393
as X, I'm sorry, as H approaches zero.

00:01:32.393 --> 00:01:34.178
I'm using that definition.

00:01:34.178 --> 00:01:35.511
Of F of X plus H

00:01:36.500 --> 00:01:38.313
minus F of X

00:01:38.313 --> 00:01:39.896
all of that over H.

00:01:41.615 --> 00:01:43.948
Well, what is F of X plus H?

00:01:44.879 --> 00:01:46.279
This is the limit

00:01:46.279 --> 00:01:47.977
as H approaches zero.

00:01:47.977 --> 00:01:49.927
F of X plus H is

00:01:49.927 --> 00:01:51.760
K times G of X plus H.

00:01:54.046 --> 00:01:55.324
Minus F of X,

00:01:55.324 --> 00:01:57.415
well that's just KG of X.

00:01:57.415 --> 00:01:58.248
KG of X.

00:01:59.759 --> 00:02:01.661
All of that over

00:02:01.661 --> 00:02:02.538
H.

00:02:02.538 --> 00:02:04.824
And then you can factor the K out.

00:02:04.824 --> 00:02:07.315
This is going to be equal to the limit

00:02:07.315 --> 00:02:09.398
as H approaches zero of K

00:02:10.725 --> 00:02:12.308
times G of X plus H

00:02:15.073 --> 00:02:16.073
minus G of X

00:02:17.342 --> 00:02:19.410
all of that over H.

00:02:19.410 --> 00:02:21.310
All I did was factor that K out.

00:02:21.310 --> 00:02:22.789
And we know from our limit properties

00:02:22.789 --> 00:02:24.668
that this is the same thing as K

00:02:24.668 --> 00:02:27.939
times the limit as H approaches zero

00:02:27.939 --> 00:02:29.772
of G of X plus H minus

00:02:31.429 --> 00:02:33.679
G of X, all of that over H.

00:02:34.735 --> 00:02:37.619
And of course, all of this
business right over there.

00:02:37.619 --> 00:02:40.188
That is just G prime of X.

00:02:40.188 --> 00:02:42.409
So, this is equal to K

00:02:42.409 --> 00:02:44.560
times G prime of X.

00:02:44.560 --> 00:02:46.672
And I know what you might be thinking.

00:02:46.672 --> 00:02:48.736
Whoa, hey this feels like it was probably

00:02:48.736 --> 00:02:51.921
going to be true, so I
just assumed it was true.

00:02:51.921 --> 00:02:53.624
But you can't just assume that.

00:02:53.624 --> 00:02:54.986
I don't know, sometimes you can

00:02:54.986 --> 00:02:56.651
when you're first trying
to get your head around it

00:02:56.651 --> 00:02:58.695
you can say, oh, this seems
like a reasonable thing.

00:02:58.695 --> 00:03:01.272
But in math, we like to
really know that it is true

00:03:01.272 --> 00:03:03.757
otherwise we will build all sorts of

00:03:03.757 --> 00:03:06.349
conclusions based on unsound foundations.

00:03:06.349 --> 00:03:08.248
This allows us to ensure
that this is something

00:03:08.248 --> 00:03:09.861
that we can do.

00:03:09.861 --> 00:03:11.952
So, this is, it's good

00:03:11.952 --> 00:03:13.735
to go through what might feel like

00:03:13.735 --> 00:03:17.097
a little bit of work to
get to this conclusion.

00:03:17.097 --> 00:03:19.953
Now let's do the third property.

00:03:19.953 --> 00:03:21.336
The third property

00:03:21.336 --> 00:03:22.253
is the idea

00:03:23.344 --> 00:03:25.298
that if I have some function

00:03:25.298 --> 00:03:26.751
that's the sum or the difference of

00:03:26.751 --> 00:03:27.996
two other functions,

00:03:27.996 --> 00:03:30.120
so G of X, and let's see,

00:03:30.120 --> 00:03:31.793
I'm using H a lot so,

00:03:31.793 --> 00:03:33.497
H, let's say,

00:03:33.497 --> 00:03:35.951
I don't know, J of X.

00:03:35.951 --> 00:03:39.215
J, yeah sure, why not, J of X.

00:03:39.215 --> 00:03:41.368
You don't see a lot of J of X's out there.

00:03:41.368 --> 00:03:43.118
Well then, well then,

00:03:44.214 --> 00:03:45.760
F prime of X

00:03:45.760 --> 00:03:47.179
is going to be equal to

00:03:47.179 --> 00:03:48.676
G prime of X

00:03:48.676 --> 00:03:50.176
plus J prime of X.

00:03:51.825 --> 00:03:53.094
And this would also have been true

00:03:53.094 --> 00:03:54.996
if this is, instead of
it being a positive here,

00:03:54.996 --> 00:03:57.130
if this was a negative.

00:03:57.130 --> 00:03:59.582
Instead of addition, if
this was subtraction.

00:03:59.582 --> 00:04:01.815
If the sum were a
difference of two functions

00:04:01.815 --> 00:04:03.248
then your derivative is gonna be the sum

00:04:03.248 --> 00:04:05.419
or the difference of their derivatives.

00:04:05.419 --> 00:04:08.197
And once again, we can
just go to the definition

00:04:08.197 --> 00:04:09.530
of F prime of X.

00:04:10.534 --> 00:04:11.867
So, F prime of X

00:04:13.222 --> 00:04:15.385
is going to be equal to the limit

00:04:15.385 --> 00:04:17.440
as H approaches zero

00:04:17.440 --> 00:04:19.498
of F of X plus H.

00:04:19.498 --> 00:04:21.366
But what is F of X plus H?

00:04:21.366 --> 00:04:23.533
Well, that's G of X plus H

00:04:26.264 --> 00:04:27.847
plus J of X plus H.

00:04:29.558 --> 00:04:31.359
So that's F of X plus H

00:04:31.359 --> 00:04:33.198
minus F of X.

00:04:33.198 --> 00:04:34.865
So F of X is G of X.

00:04:36.685 --> 00:04:38.185
G of X plus J of X

00:04:39.399 --> 00:04:41.008
plus J of X.

00:04:41.008 --> 00:04:43.158
Notice, this is F of X plus H

00:04:43.158 --> 00:04:44.176
minus F of X

00:04:44.176 --> 00:04:46.893
and we're gonna put all of that over H.

00:04:46.893 --> 00:04:49.031
So we can put all of that

00:04:49.031 --> 00:04:51.393
all of that over H.

00:04:51.393 --> 00:04:52.847
Well, what is that equal to?

00:04:52.847 --> 00:04:54.776
Well, we could just rearrange what we see

00:04:54.776 --> 00:04:56.194
on top here.

00:04:56.194 --> 00:04:57.527
This is equal to

00:04:58.559 --> 00:04:59.696
the limit

00:04:59.696 --> 00:05:01.924
as H approaches zero.

00:05:01.924 --> 00:05:04.484
Well let's see, all the mentions of G of X

00:05:04.484 --> 00:05:05.432
I'm gonna put up front.

00:05:05.432 --> 00:05:06.765
So G of X plus H

00:05:07.826 --> 00:05:08.909
minus G of X.

00:05:10.896 --> 00:05:12.396
Plus J of X plus H

00:05:15.089 --> 00:05:16.172
minus J of X.

00:05:18.208 --> 00:05:20.200
And then all of that, I could write it

00:05:20.200 --> 00:05:23.345
as this, all of that over H.

00:05:23.345 --> 00:05:26.198
Or, I could, that's the same thing as

00:05:26.198 --> 00:05:27.742
this over H

00:05:27.742 --> 00:05:29.758
plus that over H.

00:05:29.758 --> 00:05:31.014
And once again,

00:05:31.014 --> 00:05:33.691
we know from our limit properties

00:05:33.691 --> 00:05:36.292
that that is the exact same thing as

00:05:36.292 --> 00:05:38.875
the limit as H approaches zero,

00:05:40.417 --> 00:05:41.750
of G of X plus H

00:05:42.620 --> 00:05:43.620
minus G of X

00:05:44.901 --> 00:05:46.452
all of that over H

00:05:46.452 --> 00:05:49.369
plus the limit as H approaches zero

00:05:51.101 --> 00:05:52.434
of J of X plus H

00:05:54.096 --> 00:05:55.096
minus J of X

00:05:56.386 --> 00:05:57.755
all of that over H.

00:05:57.755 --> 00:05:59.085
And this, right over here,

00:05:59.085 --> 00:06:01.004
that is the definition of

00:06:01.004 --> 00:06:02.087
G prime of X.

00:06:02.997 --> 00:06:04.833
And this right over here is

00:06:04.833 --> 00:06:07.294
J prime, J prime of X

00:06:07.294 --> 00:06:08.197
and we're done.

00:06:08.197 --> 00:06:09.604
And this, instead of a positive,

00:06:09.604 --> 00:06:11.031
this was instead of addition

00:06:11.031 --> 00:06:12.599
this was subtraction.

00:06:12.599 --> 00:06:14.572
Well then that subtraction
would carry through

00:06:14.572 --> 00:06:15.972
and then instead of addition here

00:06:15.972 --> 00:06:17.541
we would have subtraction.

00:06:17.541 --> 00:06:19.025
So hopefully this makes you feel good

00:06:19.025 --> 00:06:20.469
about these properties.

00:06:20.469 --> 00:06:22.639
The properties themselves
are somewhat straightforward.

00:06:22.639 --> 00:06:25.653
You could probably guess at them.

00:06:25.653 --> 00:06:27.747
But it's nice to use the definition

00:06:27.747 --> 00:06:30.764
of our derivatives to actually

00:06:30.764 --> 00:06:34.847
feel that they are very
good conclusions to make.