Worked example: Differentiating related functions

WEBVTT
Kind: captions
Language: en

00:00:00.534 --> 00:00:03.773
- The differentiable
functions x and y are related

00:00:03.773 --> 00:00:05.965
by the following equation.

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The sine of x plus cosine of y

00:00:09.578 --> 00:00:12.479
is equal to square root of two.

00:00:12.479 --> 00:00:15.128
They also tell us that the derivative of x

00:00:15.128 --> 00:00:17.897
with respect to t is equal to five.

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They also ask us find the derivative of y

00:00:21.530 --> 00:00:25.145
with respect to t when y
is equal to pi over four

00:00:25.145 --> 00:00:29.789
and zero is less than x
is less than pi over two.

00:00:29.789 --> 00:00:32.365
So given that they are
telling us the derivative

00:00:32.365 --> 00:00:34.685
of x with respect to t and we wanna find

00:00:34.685 --> 00:00:37.134
the derivative of y with respect to t,

00:00:37.134 --> 00:00:41.301
it's a safe assumption that
both x and y are functions of t.

00:00:42.403 --> 00:00:45.751
So you could even rewrite
this equation right over here.

00:00:45.751 --> 00:00:48.584
You could rewrite it as sine of x,

00:00:50.683 --> 00:00:53.040
which is a function of t,

00:00:53.040 --> 00:00:53.957
plus cosine

00:00:55.815 --> 00:00:58.398
of y, which is a function of t,

00:00:59.499 --> 00:01:02.082
is equal to square root of two.

00:01:02.920 --> 00:01:04.542
Now, it might confuse you a little bit,

00:01:04.542 --> 00:01:06.376
you're not used to seeing x as a function

00:01:06.376 --> 00:01:08.524
of a third variable or y as a function

00:01:08.524 --> 00:01:10.068
of something other than x.

00:01:10.068 --> 00:01:11.740
But remember, x and y are just variables.

00:01:11.740 --> 00:01:15.419
This could be f of t,
and this could be g of t

00:01:15.419 --> 00:01:17.605
instead of x of t and y of t,

00:01:17.605 --> 00:01:19.503
and that might feel a
little more natural to you.

00:01:19.503 --> 00:01:23.170
But needless to say,
if we wanna find dy dt,

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what we want to do is take the derivative

00:01:26.264 --> 00:01:29.952
with respect to t of both
sides of this equation.

00:01:29.952 --> 00:01:31.357
So let's do that.

00:01:31.357 --> 00:01:33.303
So we're gonna do it
on the left-hand side,

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so it's gonna be we take
that with respect to t,

00:01:36.541 --> 00:01:38.035
derivative of that with respect to t.

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We're gonna take the derivative
of that with respect to t.

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And then we're gonna take the derivative

00:01:42.338 --> 00:01:46.547
of the right-hand side, this
constant with respect to t.

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So let's think about each of these things.

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So what is this.

00:01:51.444 --> 00:01:53.114
Let me do this in a new color.

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The stuff that I'm doing in
this aqua color right over here,

00:01:56.622 --> 00:01:58.245
how could I write that?

00:01:58.245 --> 00:02:00.405
So I'm taking the derivative
with respect to t,

00:02:00.405 --> 00:02:04.918
I have sine of something, which
is itself a function of t.

00:02:04.918 --> 00:02:07.768
So I would just apply the chain rule here.

00:02:07.768 --> 00:02:11.935
I'm first going to take the
derivative with respect to x of

00:02:13.817 --> 00:02:14.650
sine of

00:02:16.508 --> 00:02:18.714
x, I could write sine of x of t,

00:02:18.714 --> 00:02:20.881
but I'll just revert back
to the sine of x here

00:02:20.881 --> 00:02:22.365
for simplicity.

00:02:22.365 --> 00:02:25.244
And then I will then multiply
that times the derivative

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of the inside, you could
say, with respect to t

00:02:28.766 --> 00:02:32.780
times the derivative
of x with respect to t.

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This might be a little counterintuitive

00:02:34.506 --> 00:02:36.682
to how you've applied
the chain rule before

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when we only dealt with xs and ys,

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but all that's happening,
I'm taking the derivative

00:02:41.272 --> 00:02:43.565
of the outside of the sine of something

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with respect to the something,
in this case, it is x,

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and then I'm taking the
derivative of the something,

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in this case, x with respect to t.

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Well, we can do the same thing here,

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or this second term here.

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So I wanna take the
derivative with respect to y

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of, I guess you could say the outside,

00:03:04.327 --> 00:03:05.577
of cosine of y,

00:03:07.692 --> 00:03:09.206
and then I would multiply that

00:03:09.206 --> 00:03:12.873
times the derivative
of y with respect to t.

00:03:14.264 --> 00:03:17.447
And then all of that is
going to be equal to what?

00:03:17.447 --> 00:03:20.742
Well, the derivative with
respect to t of a constant,

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square root of two is a constant,

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it's not gonna change as t changes,

00:03:23.912 --> 00:03:27.385
so its derivative, its
rate of change is zero.

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All right, so now we
just have to figure out

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all of these things.

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So first of all, the
derivative with respect to x

00:03:33.681 --> 00:03:38.277
of sine of x is cosine of
x times the derivative of x

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with respect to t, I'll
just write that out here.

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The derivative of x with respect to t.

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And then we're going to have,
it's gonna be a plus here,

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the derivative of y with respect to t.

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So plus the derivative
of y with respect to t.

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I'm just flopping the order here,

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so that this goes out front.

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Now, what's the derivative of
cosine of y with respect to y?

00:03:58.372 --> 00:04:01.336
Well, that is negative sine of y.

00:04:01.336 --> 00:04:05.265
And so, actually let me
just put a sine of y here,

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then I'm gonna have a negative.

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Erase this and put a negative there.

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And that is all going to be equal to zero.

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And so what can we figure out now?

00:04:18.743 --> 00:04:21.785
They've told us that the
derivative of x with respect to t

00:04:21.785 --> 00:04:25.398
is equal to five, they tell
us that right over here.

00:04:25.398 --> 00:04:27.481
So this is equal to five.

00:04:29.088 --> 00:04:32.679
We wanna find the derivative
of y with respect to t.

00:04:32.679 --> 00:04:36.145
They tell us what y is, y is pi over four.

00:04:36.145 --> 00:04:40.312
This, y is pi over four, so
we know this is pi over four.

00:04:41.606 --> 00:04:43.791
And let's see, we have to figure out what,

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we still have two unknowns here.

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We don't know what x is and we don't know

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what the derivative of
y with respect to t is.

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This is what we need to figure out.

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So what would x be?

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What would x be when y is pi over four?

00:04:55.420 --> 00:04:56.439
Well, to figure that out,

00:04:56.439 --> 00:05:00.222
we can go back to this original
equation right over here.

00:05:00.222 --> 00:05:03.754
So when y is pi over four, you get,

00:05:03.754 --> 00:05:04.847
let me write down.

00:05:04.847 --> 00:05:05.680
Sine of x

00:05:07.391 --> 00:05:08.808
plus cosine of pi

00:05:10.716 --> 00:05:13.984
over four is equal to square root of two.

00:05:13.984 --> 00:05:15.901
Cosine of pi over four,

00:05:17.593 --> 00:05:20.936
we revert to our unit or we
think about our unit circle.

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We're in the first quadrant.

00:05:22.558 --> 00:05:24.160
If we think in degrees,
it's a 45 degree angle,

00:05:24.160 --> 00:05:28.067
that's gonna be square
root of two over two.

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And so we can subtract
square root of two over two

00:05:30.579 --> 00:05:32.852
from both sides, which is going to give us

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sine of x is equal to, well,
if you take square root of two

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over two from square root of two,

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you're taking half of it away,

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so you're gonna have half of it left.

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So square root of two over two.

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And so, what x value, when
I take the sine of it,

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and remember, where the angle,

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if we're thinking when the
unit circle is going to be

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in that first quadrant, x
is an angle in this case

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right over here.

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Well, that's going to be
once again pi over four.

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So this tells us that x
is equal to pi over four

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when y is equal to pi over four.

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And so we know that this
is pi over four as well.

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So let me just rewrite this,

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because it's getting a little bit messy.

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So we know that five times

00:06:17.521 --> 00:06:19.354
cosine of pi over four

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minus

00:06:24.268 --> 00:06:26.968
dy dt, the derivative
of y with respect to t,

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which is what we want to figure out,

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times sine of pi over four,

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is equal to zero,

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is equal to zero, and we
put some parentheses here,

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just to clarify things a little bit.

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All right, so let's see.

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Now, it's just a little bit of algebra.

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Cosine of pi over four,

00:06:46.927 --> 00:06:49.677
we already know is square
root of two over two.

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Sine of pi over four is also
square root of two over two.

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Now let's see, what if
we divide both sides

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of this equation by square
root of two over two?

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Well, what's that gonna give us?

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Well, then, this square
root of two over two

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divided by square root of two over,

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square root of two over two
divided square root of two

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over two is gonna be one.

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Square root of two over two divided

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square root of two over
two is gonna be one.

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And then zero divided by
square root of two over two

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is just still going to be zero.

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And so this whole thing simplifies to

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five times one, which is just five,

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minus the derivative
of y with respect to t

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is equal to zero,

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and so there you have it.

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You add the derivative of y
with respect to t to both sides,

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and we get the derivative of y
with respect to t is equal to

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five, when all of these
other things are true.

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When the derivative of x
with respect to t is five,

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and the derivative and y, I should say,

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is equal to pi over four.


Differentiating related functions intro

WEBVTT
Kind: captions
Language: en

00:00:00.223 --> 00:00:02.230
- [Instructor] We are told
the differentiable functions

00:00:02.230 --> 00:00:06.323
X and Y are related by
the following equation,

00:00:06.323 --> 00:00:09.388
Y is equal to square root of X.

00:00:09.388 --> 00:00:10.565
And it's interesting, they're telling us

00:00:10.565 --> 00:00:12.742
that they are both
differentiable functions,

00:00:12.742 --> 00:00:16.541
even X is a function, must be
a function of something else.

00:00:16.541 --> 00:00:18.522
Well, they tell us that
the derivative of X

00:00:18.522 --> 00:00:20.439
with respect to T is 12

00:00:21.319 --> 00:00:24.119
and they want us to
find the derivative of Y

00:00:24.119 --> 00:00:28.093
with respect to T when X is equal to nine.

00:00:28.093 --> 00:00:30.237
So, let's just make sure
we can understand this.

00:00:30.237 --> 00:00:34.214
So, they're telling us that
both X and Y are functions.

00:00:34.214 --> 00:00:36.553
Arguably they are both functions of T.

00:00:36.553 --> 00:00:38.097
Y is a function of X

00:00:38.097 --> 00:00:39.602
but then X is a function of T

00:00:39.602 --> 00:00:41.874
so Y could also be a function of T.

00:00:41.874 --> 00:00:43.668
One way to think about it is

00:00:43.668 --> 00:00:45.668
if X is equal to F of T,

00:00:47.764 --> 00:00:51.180
then Y is equal to the square root of X

00:00:51.180 --> 00:00:53.430
which would just be F of T.

00:00:55.273 --> 00:00:56.696
Another way to think about it

00:00:56.696 --> 00:00:58.722
if you took T as your input

00:00:58.722 --> 00:01:00.671
into your function F,

00:01:00.671 --> 00:01:02.838
you're going to produce X

00:01:02.838 --> 00:01:04.578
and then if you took that as your input

00:01:04.578 --> 00:01:06.995
into the square root function

00:01:08.162 --> 00:01:10.941
you are going to produce Y.

00:01:10.941 --> 00:01:14.811
So, you could just view this
as just one big box here,

00:01:14.811 --> 00:01:16.894
that Y is a function of Y

00:01:18.420 --> 00:01:21.459
but now let's actually
answer their question.

00:01:21.459 --> 00:01:24.637
To tackle it we just have
to apply the chain rule.

00:01:24.637 --> 00:01:28.316
The chain rule tells us
that the derivative of Y

00:01:28.316 --> 00:01:30.242
with respect to T

00:01:30.242 --> 00:01:33.054
is going to be equal
to the derivative of Y

00:01:33.054 --> 00:01:36.637
with respect to X times
the derivative of X

00:01:37.539 --> 00:01:39.675
with respect to T.

00:01:39.675 --> 00:01:42.769
So, let's apply it to
this particular situation.

00:01:42.769 --> 00:01:45.428
We're gonna have the derivative of Y

00:01:45.428 --> 00:01:47.239
with respect to T

00:01:47.239 --> 00:01:49.822
is equal to the derivative of Y

00:01:50.759 --> 00:01:52.503
with respect to X.

00:01:52.503 --> 00:01:53.745
Well, what's that?

00:01:53.745 --> 00:01:55.843
Well, Y is equal to the
principle root of X.

00:01:55.843 --> 00:01:57.182
You could also write this as Y

00:01:57.182 --> 00:01:59.757
is equal to X to the one half power.

00:01:59.757 --> 00:02:01.380
We could just use the power rule.

00:02:01.380 --> 00:02:03.318
The derivative of Y with respect to X

00:02:03.318 --> 00:02:07.233
is one half X to the negative one half.

00:02:07.233 --> 00:02:08.766
So, let me write that down.

00:02:08.766 --> 00:02:11.683
One half X to the negative one half

00:02:12.903 --> 00:02:16.467
and then times the derivative of X

00:02:16.467 --> 00:02:19.085
with respect to T times
the derivative of X

00:02:19.085 --> 00:02:21.356
with respect to T.

00:02:21.356 --> 00:02:24.255
Well, let's see, we wanna find
what we have here in orange,

00:02:24.255 --> 00:02:25.772
that's what the questions asks us.

00:02:25.772 --> 00:02:28.196
They tell us when X is equal to nine

00:02:28.196 --> 00:02:29.385
and the derivative of X

00:02:29.385 --> 00:02:30.986
with respect to T is equal to 12.

00:02:30.986 --> 00:02:32.685
So, we have all of the information

00:02:32.685 --> 00:02:34.604
necessary to solve for this.

00:02:34.604 --> 00:02:38.021
So, this is going to be equal to one half

00:02:39.205 --> 00:02:42.322
times nine to the negative one half,

00:02:42.322 --> 00:02:44.205
nine to the negative one half,

00:02:44.205 --> 00:02:48.372
times DX/DT, the derivative
of X with respect to T

00:02:49.501 --> 00:02:51.584
is equal to 12, times 12.

00:02:52.766 --> 00:02:55.376
So, let's see, nine to the one half

00:02:55.376 --> 00:02:56.671
would be three,

00:02:56.671 --> 00:03:00.001
nine to the negative
one half would be 1/3,

00:03:00.001 --> 00:03:01.493
so this is 1/3,

00:03:01.493 --> 00:03:05.660
so this will all simplify
to 1/2 times 1/3 is 1/6,

00:03:07.839 --> 00:03:10.122
so we could have a six in the denominator

00:03:10.122 --> 00:03:14.357
and then we are going to
have a 12 in the numerator.

00:03:14.357 --> 00:03:19.114
So, 12/6, so the derivative
of Y with respect to T

00:03:19.114 --> 00:03:20.300
when X is equal to nine

00:03:20.300 --> 00:03:23.658
and derivative of X with
respect to 12 is two.


Applying the chain rule twice

WEBVTT
Kind: captions
Language: en

00:00:00.548 --> 00:00:02.032
- [Instructor] Let's say that Y

00:00:02.032 --> 00:00:06.032
is equal to sin of X
squared to the third power,

00:00:08.122 --> 00:00:09.201
which of course we could also write

00:00:09.201 --> 00:00:12.368
as sin of X squared to the third power

00:00:15.394 --> 00:00:16.763
and what we're curious about

00:00:16.763 --> 00:00:19.603
is what is the derivative
of this with respect to X?

00:00:19.603 --> 00:00:23.770
What is DY/DX which we
could also write as Y prime?

00:00:24.967 --> 00:00:26.663
Well, there's a couple of
ways to think about it.

00:00:26.663 --> 00:00:29.967
This isn't a straightforward
expression here

00:00:29.967 --> 00:00:32.846
but you might notice that I have something

00:00:32.846 --> 00:00:34.955
being raised to the third power,

00:00:34.955 --> 00:00:37.051
in fact, if we look at the
outside of this expression

00:00:37.051 --> 00:00:38.705
we have some business in here

00:00:38.705 --> 00:00:41.097
that's being raised to the third power.

00:00:41.097 --> 00:00:42.658
And so, one way to tackle this

00:00:42.658 --> 00:00:44.468
is to apply the chain rule.

00:00:44.468 --> 00:00:45.793
So, if we apply the chain rule

00:00:45.793 --> 00:00:47.568
it's gonna be the
derivative of the outside

00:00:47.568 --> 00:00:49.225
with respect to the inside

00:00:49.225 --> 00:00:51.373
or the something to the third power,

00:00:51.373 --> 00:00:53.219
the derivative of the
something to the third power

00:00:53.219 --> 00:00:55.315
with respect to that something.

00:00:55.315 --> 00:00:59.482
So, it's going to be three
times that something squared

00:01:00.986 --> 00:01:04.421
times the derivative with respect to X

00:01:04.421 --> 00:01:07.088
of that something, in this case,

00:01:09.212 --> 00:01:11.628
the something is sin,

00:01:11.628 --> 00:01:13.837
let me write that in the blue color,

00:01:13.837 --> 00:01:15.754
it is sin of X squared.

00:01:18.238 --> 00:01:20.155
It is sin of X squared.

00:01:23.466 --> 00:01:26.679
No matter what was inside
of these orange parentheses

00:01:26.679 --> 00:01:29.296
I would put it inside of
the orange parentheses

00:01:29.296 --> 00:01:31.554
and these orange brackets right over here.

00:01:31.554 --> 00:01:34.074
We learned that in the chain rule.

00:01:34.074 --> 00:01:36.410
So, let's see, we know
this is just a matter

00:01:36.410 --> 00:01:37.730
of the first part of the expression

00:01:37.730 --> 00:01:40.835
is just a matter of
algebraic simplification

00:01:40.835 --> 00:01:44.082
but the second part we need
to now take the derivative

00:01:44.082 --> 00:01:46.108
of sin of X squared.

00:01:46.108 --> 00:01:49.886
Well, now we would want to
use the chain rule again.

00:01:49.886 --> 00:01:51.400
So, I'm going to take the derivative,

00:01:51.400 --> 00:01:53.233
it's sin of something,

00:01:54.770 --> 00:01:57.061
so this is going to be,
the derivative of this

00:01:57.061 --> 00:01:58.138
is gonna be the sin of something

00:01:58.138 --> 00:01:59.522
with respect to something,

00:01:59.522 --> 00:02:02.439
so that is cosine of that something

00:02:03.481 --> 00:02:06.488
times the derivative with respect to X

00:02:06.488 --> 00:02:08.538
of the something.

00:02:08.538 --> 00:02:12.205
In this case, the
something is our X squared

00:02:13.041 --> 00:02:15.413
and of course, we have
all of this out front

00:02:15.413 --> 00:02:17.746
which is the three times sin

00:02:19.104 --> 00:02:23.934
of X squared, I could write
it like this, squared.

00:02:23.934 --> 00:02:25.491
Alright, so we're getting close.

00:02:25.491 --> 00:02:26.700
Now we just have to
figure out the derivative

00:02:26.700 --> 00:02:29.086
with respect to X of X squared

00:02:29.086 --> 00:02:31.746
and we've seen that many times before.

00:02:31.746 --> 00:02:33.140
That, we just use the power rule,

00:02:33.140 --> 00:02:35.227
that's going to be two X.

00:02:35.227 --> 00:02:39.394
Two X and so, if we
wanted to write the DY/DX,

00:02:40.873 --> 00:02:43.015
let me get a little bit
of a mini drum roll here,

00:02:43.015 --> 00:02:44.538
this shouldn't take us too long,

00:02:44.538 --> 00:02:48.520
DY/DX, I'll multiply the
three times the two X

00:02:48.520 --> 00:02:50.385
which is going to be six X,

00:02:50.385 --> 00:02:52.359
so I've covered those so far

00:02:52.359 --> 00:02:54.942
times sin squared of X squared,

00:02:56.899 --> 00:02:59.482
times sin squared of X squared,

00:03:00.452 --> 00:03:02.619
times cosine of X squared.

00:03:03.737 --> 00:03:07.904
And we are done applying the
chain rule multiple times.


Applying the chain rule and product rule

WEBVTT
Kind: captions
Language: en

00:00:00.658 --> 00:00:02.324
- [Instructor] What we're
going to do in this video

00:00:02.324 --> 00:00:05.268
is try to find the
derivative with respect to X

00:00:05.268 --> 00:00:06.268
of X squared

00:00:07.925 --> 00:00:08.758
sin of X.

00:00:10.068 --> 00:00:12.359
All of that to the third power.

00:00:12.359 --> 00:00:13.882
And what's going to be
interesting is that there's

00:00:13.882 --> 00:00:15.505
multiple ways to tackle it.

00:00:15.505 --> 00:00:16.925
And I encourage you to pause the video

00:00:16.925 --> 00:00:20.341
and see if you could work
through it on your own.

00:00:20.341 --> 00:00:22.682
So there's actually multiple techniques.

00:00:22.682 --> 00:00:25.504
One path is to do the chain rule first.

00:00:25.504 --> 00:00:27.087
So I'll just say CR

00:00:28.464 --> 00:00:30.215
for chain rule first.

00:00:30.215 --> 00:00:33.005
And so I have, I'm taking the
derivative with respect to X

00:00:33.005 --> 00:00:35.442
of something to the third power.

00:00:35.442 --> 00:00:37.963
So, if I take the derivative it would be

00:00:37.963 --> 00:00:40.329
the derivative with
respect to that something.

00:00:40.329 --> 00:00:41.556
So that would be

00:00:41.556 --> 00:00:43.877
three times that something

00:00:43.877 --> 00:00:45.404
squared

00:00:45.404 --> 00:00:48.826
times the derivative with respect to X

00:00:48.826 --> 00:00:50.936
of that something.

00:00:50.936 --> 00:00:52.657
Where the something, in this case,

00:00:52.657 --> 00:00:54.490
is X squared sin of X.

00:00:56.901 --> 00:00:57.984
X squared sin

00:00:59.054 --> 00:00:59.932
of X.

00:00:59.932 --> 00:01:03.244
This is just an application
of the chain rule.

00:01:03.244 --> 00:01:06.143
Now, the second part, what would that be?

00:01:06.143 --> 00:01:07.707
The second part here

00:01:07.707 --> 00:01:09.518
do this in another color.

00:01:09.518 --> 00:01:10.884
In orange.

00:01:10.884 --> 00:01:13.225
Well here, I would apply the product rule.

00:01:13.225 --> 00:01:15.325
I have a product of two expressions.

00:01:15.325 --> 00:01:17.694
So I would take the derivative
of, let me write this down.

00:01:17.694 --> 00:01:20.913
So this is gonna be the product rule.

00:01:20.913 --> 00:01:21.996
Product rule.

00:01:22.983 --> 00:01:26.179
I would take the derivative
of the first expression.

00:01:26.179 --> 00:01:29.103
So, X, derivative of X squared

00:01:29.103 --> 00:01:30.544
is two X.

00:01:30.544 --> 00:01:32.280
Let me write a little bit to the right.

00:01:32.280 --> 00:01:33.437
This is gonna be two X

00:01:33.437 --> 00:01:34.873
times the second expression

00:01:34.873 --> 00:01:35.706
sin of X.

00:01:36.923 --> 00:01:39.795
Plus the first expression X squared

00:01:39.795 --> 00:01:41.695
times the derivative of the second one.

00:01:41.695 --> 00:01:42.662
Cosin of X.

00:01:42.662 --> 00:01:43.862
That's just the product rule

00:01:43.862 --> 00:01:45.685
as applied to this part

00:01:45.685 --> 00:01:47.292
right over here.

00:01:47.292 --> 00:01:49.289
And all of that, of course,
is being multiplied by

00:01:49.289 --> 00:01:50.802
this up front.

00:01:50.802 --> 00:01:52.460
Which actually, let me just

00:01:52.460 --> 00:01:54.120
rewrite that.

00:01:54.120 --> 00:01:57.581
So all of this I could rewrite as

00:01:57.581 --> 00:01:59.532
let's see, this would be

00:01:59.532 --> 00:02:00.449
three times

00:02:02.647 --> 00:02:04.848
if I have the product
raised to the second power

00:02:04.848 --> 00:02:06.500
I could take each of
them to the second power

00:02:06.500 --> 00:02:07.642
and then take their products.

00:02:07.642 --> 00:02:09.558
So X squared squared is

00:02:09.558 --> 00:02:11.321
X to the fourth.

00:02:11.321 --> 00:02:13.019
And then sin of X squared is

00:02:13.019 --> 00:02:14.415
sin

00:02:14.415 --> 00:02:16.023
squared of X.

00:02:16.023 --> 00:02:18.931
And then all of that is
being multiplied by that.

00:02:18.931 --> 00:02:22.151
And, if we want, we can
algebraically simplify.

00:02:22.151 --> 00:02:24.835
We can distribute everything out.

00:02:24.835 --> 00:02:27.581
In which case, what would we get?

00:02:27.581 --> 00:02:28.556
Well let's see.

00:02:28.556 --> 00:02:30.473
Three times two is six.

00:02:31.718 --> 00:02:33.698
X to the fourth times X is

00:02:33.698 --> 00:02:35.011
X to the fifth.

00:02:35.011 --> 00:02:36.856
Sin squared of X times sin of X

00:02:36.856 --> 00:02:37.773
is sin of X

00:02:38.910 --> 00:02:40.432
to the third power.

00:02:40.432 --> 00:02:42.595
And then, let's see, three

00:02:42.595 --> 00:02:45.742
so plus three, X to the
fourth times X squared is

00:02:45.742 --> 00:02:47.465
X to the sixth power.

00:02:47.465 --> 00:02:50.132
And then I have sin squared of X

00:02:50.132 --> 00:02:51.770
sin squared of X

00:02:51.770 --> 00:02:52.687
cosin of X.

00:02:53.528 --> 00:02:55.257
So there you have it, that's one strategy

00:02:55.257 --> 00:02:57.539
chain rule first, and then product rule.

00:02:57.539 --> 00:02:59.448
What would be another strategy

00:02:59.448 --> 00:03:02.098
pause the video and try to think of it.

00:03:02.098 --> 00:03:04.250
Well, we could just algebraically

00:03:04.250 --> 00:03:06.520
use our exponent properties first.

00:03:06.520 --> 00:03:09.313
In which case, this is
going to be equal to

00:03:09.313 --> 00:03:11.954
the derivative with respect to X

00:03:11.954 --> 00:03:12.787
of

00:03:14.618 --> 00:03:17.585
if I'm taking X squared times
sin of X to the third power

00:03:17.585 --> 00:03:19.584
instead I could say X to the third power

00:03:19.584 --> 00:03:22.986
which is going to be X to the sixth.

00:03:22.986 --> 00:03:24.860
And then sin of X to the third power.

00:03:24.860 --> 00:03:25.693
Sin of X

00:03:26.840 --> 00:03:27.947
to the

00:03:27.947 --> 00:03:28.849
third power.

00:03:28.849 --> 00:03:30.929
I'm using the same exponent property

00:03:30.929 --> 00:03:34.255
that we used right over
here to simplify this.

00:03:34.255 --> 00:03:36.334
If I have if I'm doing the product things

00:03:36.334 --> 00:03:38.396
to some exponent, well
that's the same thing

00:03:38.396 --> 00:03:40.012
if each of them raised to the exponent

00:03:40.012 --> 00:03:42.036
and then the product of the two.

00:03:42.036 --> 00:03:43.919
Now how would we tackle this?

00:03:43.919 --> 00:03:45.867
Well, I here, I would do

00:03:45.867 --> 00:03:47.970
the product rule first.

00:03:47.970 --> 00:03:49.337
So let's do that.

00:03:49.337 --> 00:03:51.379
So let's do the product

00:03:51.379 --> 00:03:52.638
rule.

00:03:52.638 --> 00:03:53.904
So, we're gonna take

00:03:53.904 --> 00:03:56.125
the derivative of the first expression.

00:03:56.125 --> 00:03:57.619
So, derivative of X to the sixth

00:03:57.619 --> 00:03:58.452
is six

00:03:59.508 --> 00:04:01.568
X to the fifth.

00:04:01.568 --> 00:04:03.053
Times the second expression.

00:04:03.053 --> 00:04:04.284
Sin

00:04:04.284 --> 00:04:06.076
to the third of X.

00:04:06.076 --> 00:04:07.921
Or, sin of X to the third power.

00:04:07.921 --> 00:04:10.754
Plus the first X to the sixth

00:04:10.754 --> 00:04:12.376
times the derivative of the second

00:04:12.376 --> 00:04:13.937
and I'm just gonna write that

00:04:13.937 --> 00:04:14.976
D DX of

00:04:14.976 --> 00:04:15.809
sin of X

00:04:17.576 --> 00:04:19.159
to the third power.

00:04:20.193 --> 00:04:23.342
Now, to evaluate this right over here

00:04:23.342 --> 00:04:26.125
it does definitely make sense

00:04:26.125 --> 00:04:28.650
to use the chain rule.

00:04:28.650 --> 00:04:30.298
Use

00:04:30.298 --> 00:04:31.695
the chain rule.

00:04:31.695 --> 00:04:34.224
And so what is this going to be?

00:04:34.224 --> 00:04:36.057
Well I have the derivative of something

00:04:36.057 --> 00:04:37.069
to the third power.

00:04:37.069 --> 00:04:39.086
So that's going to be three

00:04:39.086 --> 00:04:40.777
times that something squared

00:04:40.777 --> 00:04:43.751
times the derivative of that something.

00:04:43.751 --> 00:04:46.678
So in this case, the
something is sin of X.

00:04:46.678 --> 00:04:49.867
And the derivative of
sin of X is cosin of X.

00:04:49.867 --> 00:04:51.987
Then I have all of this
business over here.

00:04:51.987 --> 00:04:54.823
I have six X to the fifth.

00:04:54.823 --> 00:04:56.647
Sin to the third

00:04:56.647 --> 00:04:58.422
or sin of X to the third power.

00:04:58.422 --> 00:04:59.472
Plus

00:04:59.472 --> 00:05:00.722
X to the sixth.

00:05:02.179 --> 00:05:04.331
And if I were to just
simplify this a little bit

00:05:04.331 --> 00:05:05.798
in fact, you see it very clearly.

00:05:05.798 --> 00:05:08.467
These two things are equivalent.

00:05:08.467 --> 00:05:09.300
This

00:05:10.151 --> 00:05:12.402
this term is exactly
equivalent to this term

00:05:12.402 --> 00:05:13.413
the way it's written.

00:05:13.413 --> 00:05:15.545
And then this is exactly,

00:05:15.545 --> 00:05:18.399
if you multiply three times X to the sixth

00:05:18.399 --> 00:05:20.123
sin of X squared

00:05:20.123 --> 00:05:21.547
cosin of X.

00:05:21.547 --> 00:05:23.143
So, the nice thing about math

00:05:23.143 --> 00:05:24.982
if we're doing things
that make logical sense

00:05:24.982 --> 00:05:26.842
we should get to the same endpoint.

00:05:26.842 --> 00:05:29.583
But the point here is that
there's multiple strategies.

00:05:29.583 --> 00:05:30.940
You could use a chain rule first

00:05:30.940 --> 00:05:32.077
and then the product rule.

00:05:32.077 --> 00:05:33.378
Or you could use a product rule first,

00:05:33.378 --> 00:05:34.502
and then the chain rule.

00:05:34.502 --> 00:05:36.910
In this case, you could
debate which one is faster.

00:05:36.910 --> 00:05:38.040
It looks like the one on the right

00:05:38.040 --> 00:05:39.886
might be a little bit faster.

00:05:39.886 --> 00:05:40.956
But

00:05:40.956 --> 00:05:42.060
sometimes

00:05:42.060 --> 00:05:43.533
these two are pretty close.

00:05:43.533 --> 00:05:45.640
But sometimes it'll be more clear than not

00:05:45.640 --> 00:05:47.428
which one is preferable.

00:05:47.428 --> 00:05:49.588
You really want to minimize
the amount of hairiness,

00:05:49.588 --> 00:05:51.355
the amount the number of steps

00:05:51.355 --> 00:05:53.659
the chances for careless
mistakes you might have.


Quotient rule

WEBVTT
Kind: captions
Language: en

00:00:00.656 --> 00:00:02.502
- [Instructor] What we're
going to do in this video

00:00:02.502 --> 00:00:03.920
is introduce ourselves

00:00:03.920 --> 00:00:05.670
to the quotient rule.

00:00:06.535 --> 00:00:08.854
And we're not going to
prove it in this video.

00:00:08.854 --> 00:00:10.837
In a future video we can prove
it using the product rule

00:00:10.837 --> 00:00:13.581
and we'll see it has some
similarities to the product rule.

00:00:13.581 --> 00:00:15.681
But here, we'll learn about what it is

00:00:15.681 --> 00:00:19.032
and how and where to actually apply it.

00:00:19.032 --> 00:00:20.639
So for example

00:00:20.639 --> 00:00:22.803
if I have some function F of X

00:00:22.803 --> 00:00:25.019
and it can be expressed as the quotient

00:00:25.019 --> 00:00:26.125
of two expressions.

00:00:26.125 --> 00:00:27.324
So let's say

00:00:27.324 --> 00:00:28.157
U of X

00:00:29.074 --> 00:00:30.074
over V of X.

00:00:32.196 --> 00:00:34.233
Then the quotient rule tells us

00:00:34.233 --> 00:00:36.374
that F prime of X

00:00:36.374 --> 00:00:38.101
is going to be equal to

00:00:38.101 --> 00:00:39.753
and this is going to look
a little bit complicated

00:00:39.753 --> 00:00:41.496
but once we apply it, you'll hopefully

00:00:41.496 --> 00:00:43.320
get a little bit more comfortable with it.

00:00:43.320 --> 00:00:45.041
Its going to be equal to the derivative

00:00:45.041 --> 00:00:46.635
of the numerator function.

00:00:46.635 --> 00:00:48.404
U prime of X.

00:00:48.404 --> 00:00:51.051
Times the denominator function.

00:00:51.051 --> 00:00:52.292
V of X.

00:00:52.292 --> 00:00:53.125
Minus

00:00:54.437 --> 00:00:56.482
the numerator function.

00:00:56.482 --> 00:00:57.315
U of X.

00:00:58.874 --> 00:01:00.654
Do that in that blue color.

00:01:00.654 --> 00:01:01.487
U of X.

00:01:02.535 --> 00:01:04.870
Times the derivative of
the denominator function

00:01:04.870 --> 00:01:07.168
times V prime of X.

00:01:07.168 --> 00:01:09.913
And this already looks very
similar to the product rule.

00:01:09.913 --> 00:01:11.998
If this was U of X times V of X

00:01:11.998 --> 00:01:14.040
then this is what we would
get if we took the derivative

00:01:14.040 --> 00:01:15.258
this was a plus sign.

00:01:15.258 --> 00:01:16.809
But this is here, a minus sign.

00:01:16.809 --> 00:01:17.900
But were not done yet.

00:01:17.900 --> 00:01:19.817
We would then divide by

00:01:21.189 --> 00:01:23.723
the denominator function squared.

00:01:23.723 --> 00:01:24.556
V of X

00:01:26.646 --> 00:01:27.880
squared.

00:01:27.880 --> 00:01:31.510
So let's actually apply this idea.

00:01:31.510 --> 00:01:34.659
So let's say that we have F of X

00:01:34.659 --> 00:01:36.393
is equal to

00:01:36.393 --> 00:01:37.646
X squared

00:01:37.646 --> 00:01:39.063
over cosine of X.

00:01:40.739 --> 00:01:42.371
Well what could be our U of X

00:01:42.371 --> 00:01:43.774
and what could be our V of X?

00:01:43.774 --> 00:01:44.902
Well, our U of X

00:01:44.902 --> 00:01:46.417
could be our X squared.

00:01:46.417 --> 00:01:47.834
So that is U of X

00:01:48.990 --> 00:01:52.407
and U prime of X would be equal to two X.

00:01:53.415 --> 00:01:56.243
And then this could be our V of X.

00:01:56.243 --> 00:01:57.743
So this is V of X.

00:01:58.954 --> 00:02:00.961
And V prime of X.

00:02:00.961 --> 00:02:03.117
The derivative of cosine
of X with respect to X

00:02:03.117 --> 00:02:06.171
is equal to negative sine of X.

00:02:06.171 --> 00:02:07.844
And then we just apply this.

00:02:07.844 --> 00:02:09.154
So based on that

00:02:09.154 --> 00:02:10.154
F prime of X

00:02:11.336 --> 00:02:12.726
is going to be equal to

00:02:12.726 --> 00:02:15.156
the derivative of the numerator function

00:02:15.156 --> 00:02:18.353
that's two X, right over
here, that's that there.

00:02:18.353 --> 00:02:19.639
So it's gonna be two X

00:02:19.639 --> 00:02:21.591
times the denominator function.

00:02:21.591 --> 00:02:23.972
V of X is just cosine of X

00:02:23.972 --> 00:02:25.472
times cosine of X.

00:02:26.473 --> 00:02:27.306
Minus

00:02:28.281 --> 00:02:29.676
the numerator function

00:02:29.676 --> 00:02:32.014
which is just X squared.

00:02:32.014 --> 00:02:33.605
X squared.

00:02:33.605 --> 00:02:36.518
Times the derivative of
the denominator function.

00:02:36.518 --> 00:02:37.935
The derivative of cosine of X

00:02:37.935 --> 00:02:39.561
is negative sine X.

00:02:39.561 --> 00:02:40.918
So, negative

00:02:40.918 --> 00:02:41.751
sine of X.

00:02:43.377 --> 00:02:45.240
All of that over

00:02:45.240 --> 00:02:46.573
all of that over

00:02:47.508 --> 00:02:49.902
the denominator function squared.

00:02:49.902 --> 00:02:52.256
So that's cosine of X

00:02:52.256 --> 00:02:54.070
and I'm going to square it.

00:02:54.070 --> 00:02:55.963
I could write it, of course, like this.

00:02:55.963 --> 00:02:57.320
Actually, let me write it like that

00:02:57.320 --> 00:02:58.678
just to make it a little bit clearer.

00:02:58.678 --> 00:03:01.362
And at this point, we
just have to simplify.

00:03:01.362 --> 00:03:04.134
This is going to be equal to

00:03:04.134 --> 00:03:05.785
let's see, we're gonna get

00:03:05.785 --> 00:03:07.785
two X times cosine of X.

00:03:08.890 --> 00:03:10.377
Two X

00:03:10.377 --> 00:03:11.377
cosine of X.

00:03:12.334 --> 00:03:15.328
Negative times a negative is a positive.

00:03:15.328 --> 00:03:16.578
Plus, X squared

00:03:18.084 --> 00:03:19.639
X squared

00:03:19.639 --> 00:03:20.972
times sine of X.

00:03:21.913 --> 00:03:23.297
Sine

00:03:23.297 --> 00:03:24.304
of X.

00:03:24.304 --> 00:03:25.637
All of that over

00:03:27.071 --> 00:03:29.310
cosine of X squared.

00:03:29.310 --> 00:03:31.280
Which I could write like this, as well.

00:03:31.280 --> 00:03:32.414
And we're done.

00:03:32.414 --> 00:03:33.826
You could try to simplify it, in fact,

00:03:33.826 --> 00:03:36.654
there's not an obvious way
to simplify this any further.

00:03:36.654 --> 00:03:38.257
Now what you'll see in the future

00:03:38.257 --> 00:03:39.662
you might already know something called

00:03:39.662 --> 00:03:42.000
the chain rule, or you might
learn it in the future.

00:03:42.000 --> 00:03:43.749
But you could also do the quotient rule

00:03:43.749 --> 00:03:45.529
using the product and the chain rule

00:03:45.529 --> 00:03:47.167
that you might learn in the future.

00:03:47.167 --> 00:03:48.558
But if you don't know the chain rule yet,

00:03:48.558 --> 00:03:50.475
this is fairly useful.


Proof: the derivative of ln(x) is 1/x

WEBVTT
Kind: captions
Language: en

00:00:00.821 --> 00:00:02.792
- [Instructor] What we're
going to do in this video

00:00:02.792 --> 00:00:05.055
is prove to ourselves that the derivative

00:00:05.055 --> 00:00:07.006
with respect to X

00:00:07.006 --> 00:00:08.589
of natural log of X

00:00:09.423 --> 00:00:11.367
is indeed, equal to

00:00:11.367 --> 00:00:12.284
one over X.

00:00:13.259 --> 00:00:15.019
So let's get started.

00:00:15.019 --> 00:00:17.436
So just using the
definition of a derivative

00:00:17.436 --> 00:00:19.090
if I were to say the derivative

00:00:19.090 --> 00:00:20.321
with respect to X

00:00:20.321 --> 00:00:21.904
of natural log of X

00:00:22.984 --> 00:00:25.493
that is gonna be the limit

00:00:25.493 --> 00:00:27.660
as delta X approaches zero

00:00:28.587 --> 00:00:31.587
of the natural log of X plus delta X

00:00:33.496 --> 00:00:35.746
minus the natural log of X.

00:00:36.585 --> 00:00:37.918
All of that over

00:00:39.019 --> 00:00:40.516
delta X.

00:00:40.516 --> 00:00:43.671
So now we can use a few
logarithm properties.

00:00:43.671 --> 00:00:46.146
We know that if I have
the natural log of A

00:00:46.146 --> 00:00:48.730
minus the natural log of B

00:00:48.730 --> 00:00:50.832
this is equal to the natural log

00:00:50.832 --> 00:00:52.649
of A over B.

00:00:52.649 --> 00:00:54.360
So we can use that right over here

00:00:54.360 --> 00:00:56.319
where I have the natural
log over something

00:00:56.319 --> 00:00:58.059
minus the natural log of something else.

00:00:58.059 --> 00:01:00.186
So all of this is going to be equal to

00:01:00.186 --> 00:01:01.853
the limit as delta X

00:01:03.272 --> 00:01:04.841
approaches zero

00:01:04.841 --> 00:01:06.341
of the natural log

00:01:07.481 --> 00:01:09.481
of this divided by that.

00:01:10.405 --> 00:01:12.085
So X plus delta X

00:01:12.085 --> 00:01:12.918
over X.

00:01:14.367 --> 00:01:15.534
X plus delta X

00:01:17.305 --> 00:01:18.799
over X

00:01:18.799 --> 00:01:20.129
all of that over

00:01:20.129 --> 00:01:21.582
delta X.

00:01:21.582 --> 00:01:23.272
And actually, let me
just write it that way.

00:01:23.272 --> 00:01:24.417
All of that over

00:01:24.417 --> 00:01:25.585
delta X.

00:01:25.585 --> 00:01:27.325
Well jee, all I did, if I have

00:01:27.325 --> 00:01:29.351
natural log of that
minus natural log of that

00:01:29.351 --> 00:01:31.338
that's the same thing as natural log

00:01:31.338 --> 00:01:32.723
of this first expression

00:01:32.723 --> 00:01:34.585
divided by that second expression.

00:01:34.585 --> 00:01:36.907
It comes straight out of
our logarithm properties.

00:01:36.907 --> 00:01:41.248
Well, inside of this logarithm
X divided by X is one.

00:01:41.248 --> 00:01:43.236
And then delta X divided by X

00:01:43.236 --> 00:01:46.026
just write that as delta X

00:01:46.026 --> 00:01:46.859
over X.

00:01:47.945 --> 00:01:49.590
So that's another way of writing that.

00:01:49.590 --> 00:01:52.366
And then we could put a
one over delta X up front.

00:01:52.366 --> 00:01:55.780
So we could say this is the same thing as

00:01:55.780 --> 00:01:57.666
this is equal to

00:01:57.666 --> 00:01:58.499
the limit

00:01:59.342 --> 00:02:00.175
as delta X

00:02:01.164 --> 00:02:02.390
approaches zero

00:02:02.390 --> 00:02:05.028
of, I 'll do this in another color.

00:02:05.028 --> 00:02:09.152
So this, I can rewrite as one over delta X

00:02:09.152 --> 00:02:11.627
times the natural log

00:02:11.627 --> 00:02:12.460
of

00:02:16.122 --> 00:02:17.742
one plus

00:02:17.742 --> 00:02:19.075
one plus delta X

00:02:20.469 --> 00:02:21.872
over X.

00:02:21.872 --> 00:02:24.200
Let me close that parenthesis.

00:02:24.200 --> 00:02:27.935
So now we can use another
exponent property.

00:02:27.935 --> 00:02:30.989
If I have, I'll write it out here.

00:02:30.989 --> 00:02:34.024
If I have A times a natural log of B

00:02:34.024 --> 00:02:36.369
that is equivalent to

00:02:36.369 --> 00:02:38.587
the natural log of B

00:02:38.587 --> 00:02:39.420
to the A.

00:02:40.265 --> 00:02:42.836
And so here, this would
be the A in that case.

00:02:42.836 --> 00:02:46.070
So I could bring that out and
make that an exponent on this.

00:02:46.070 --> 00:02:48.622
So this is all going to be equal to

00:02:48.622 --> 00:02:50.235
the limit

00:02:50.235 --> 00:02:53.154
as delta X approaches zero

00:02:53.154 --> 00:02:54.955
of the natural log

00:02:54.955 --> 00:02:57.205
of, give myself some space,

00:02:58.492 --> 00:02:59.325
one

00:03:02.587 --> 00:03:03.920
one plus delta X

00:03:05.407 --> 00:03:06.240
over X

00:03:07.198 --> 00:03:08.233
to the

00:03:08.233 --> 00:03:10.238
one over delta X power.

00:03:10.238 --> 00:03:12.708
One over delta X power.

00:03:12.708 --> 00:03:15.396
Now this might start to
look familiar to you,

00:03:15.396 --> 00:03:18.049
it might start to look close
to the definition of E.

00:03:18.049 --> 00:03:20.326
And we are, indeed, getting close to that.

00:03:20.326 --> 00:03:21.908
In order to get there fully

00:03:21.908 --> 00:03:24.366
I'm gonna do a change of variable.

00:03:24.366 --> 00:03:25.951
I'm going to say

00:03:25.951 --> 00:03:26.784
let's let

00:03:27.700 --> 00:03:28.533
let's let

00:03:29.657 --> 00:03:31.574
N equal delta X over X.

00:03:33.572 --> 00:03:34.923
Delta X

00:03:34.923 --> 00:03:36.222
over X.

00:03:36.222 --> 00:03:37.839
Well, in that case,

00:03:37.839 --> 00:03:40.196
then if you multiply both sides by X

00:03:40.196 --> 00:03:42.779
you get delta X is equal to NX.

00:03:43.663 --> 00:03:46.066
Again, just multiply both
sides of this equation by X

00:03:46.066 --> 00:03:48.434
and swap to the sides there.

00:03:48.434 --> 00:03:51.404
And then if you want one over delta X

00:03:51.404 --> 00:03:52.737
one over delta X

00:03:53.933 --> 00:03:55.727
that would be equal to

00:03:55.727 --> 00:03:56.644
one over NX

00:03:57.918 --> 00:04:00.487
which we could also write as one over N

00:04:00.487 --> 00:04:01.904
times one over X.

00:04:02.977 --> 00:04:04.359
Or actually, let me write it this way.

00:04:04.359 --> 00:04:06.098
Actually, that's the way
I do want to write it.

00:04:06.098 --> 00:04:08.259
So these are all the substitutions

00:04:08.259 --> 00:04:10.256
that I want to do when
I change a variable.

00:04:10.256 --> 00:04:12.003
And we also want to say, well look.

00:04:12.003 --> 00:04:14.503
As delta X is approaching zero

00:04:15.495 --> 00:04:17.953
what is N going to approach?

00:04:17.953 --> 00:04:20.620
Well, as delta X approaches zero

00:04:22.554 --> 00:04:24.798
we have N will approach zero, as well.

00:04:24.798 --> 00:04:27.094
Zero over X, well that's
just gonna be zero

00:04:27.094 --> 00:04:30.227
for any X that is not equal to zero.

00:04:30.227 --> 00:04:31.912
And that's okay because

00:04:31.912 --> 00:04:33.535
zero is not even in the domain

00:04:33.535 --> 00:04:34.954
of the natural log of X.

00:04:34.954 --> 00:04:38.260
So, this is going to be,
for our domain, it works.

00:04:38.260 --> 00:04:39.860
That is, delta X approaches zero

00:04:39.860 --> 00:04:42.088
N approaches zero.

00:04:42.088 --> 00:04:44.872
And you could think about
it the other way around

00:04:44.872 --> 00:04:47.403
as N approaches zero,
delta X approaches zero.

00:04:47.403 --> 00:04:50.455
So now let's do our change of variable.

00:04:50.455 --> 00:04:52.719
So if we make the substitutions

00:04:52.719 --> 00:04:55.438
instead of taking the limit
as delta X approaches zero

00:04:55.438 --> 00:04:57.112
we are now going to take the limit as

00:04:57.112 --> 00:04:58.529
N approaches zero

00:05:00.106 --> 00:05:01.617
of

00:05:01.617 --> 00:05:03.243
the natural log

00:05:03.243 --> 00:05:04.847
of

00:05:04.847 --> 00:05:06.796
give myself some parenthesis.

00:05:06.796 --> 00:05:08.629
And I'll say, one plus

00:05:09.540 --> 00:05:12.442
and now this is the same thing as N.

00:05:12.442 --> 00:05:13.766
One plus N.

00:05:13.766 --> 00:05:15.786
And then all of that is going to be

00:05:15.786 --> 00:05:17.129
raised to

00:05:17.129 --> 00:05:19.020
the one over N times one over X

00:05:19.020 --> 00:05:21.386
that's what one over delta X is equal to.

00:05:21.386 --> 00:05:23.756
This is one over delta X right over here.

00:05:23.756 --> 00:05:25.037
Which we have over here.

00:05:25.037 --> 00:05:26.573
And it's the same thing as one over N

00:05:26.573 --> 00:05:27.874
times one over X.

00:05:27.874 --> 00:05:29.169
So let me write that down.

00:05:29.169 --> 00:05:31.007
So this is the same thing

00:05:31.007 --> 00:05:32.590
as one over N times

00:05:33.498 --> 00:05:34.415
one over X.

00:05:36.606 --> 00:05:39.224
Now we can use this same exponent property

00:05:39.224 --> 00:05:40.538
to go the other way around.

00:05:40.538 --> 00:05:42.699
Well actually, let me just
rewrite this another time.

00:05:42.699 --> 00:05:44.968
So this is going to be the same thing

00:05:44.968 --> 00:05:45.968
as the limit

00:05:47.186 --> 00:05:48.853
as N approaches zero

00:05:50.155 --> 00:05:52.479
of the natural log

00:05:52.479 --> 00:05:53.312
of

00:05:55.299 --> 00:05:57.946
one plus, I'll just write this in orange.

00:05:57.946 --> 00:05:59.201
One plus N

00:05:59.201 --> 00:06:01.466
to the one over N.

00:06:01.466 --> 00:06:02.864
If I raise something to the exponent

00:06:02.864 --> 00:06:04.338
and that's times something else,

00:06:04.338 --> 00:06:05.991
I can that's the same thing as raising

00:06:05.991 --> 00:06:07.199
it to the first exponent.

00:06:07.199 --> 00:06:09.480
And then raising that to the second value.

00:06:09.480 --> 00:06:11.849
This comes, once again,
straight out of our

00:06:11.849 --> 00:06:13.838
exponent properties.

00:06:13.838 --> 00:06:16.971
And now we can use this
property the other way

00:06:16.971 --> 00:06:19.493
to bring this one over X out front.

00:06:19.493 --> 00:06:22.277
But in fact, the one over
X itself is not affected

00:06:22.277 --> 00:06:23.783
as N approaches zero.

00:06:23.783 --> 00:06:26.421
So we can even take it
completely out of the limit.

00:06:26.421 --> 00:06:29.113
So we could take it
all the way over there.

00:06:29.113 --> 00:06:31.535
And this is when you
should be getting excited.

00:06:31.535 --> 00:06:33.419
This will be equal to

00:06:33.419 --> 00:06:34.252
one over X

00:06:35.555 --> 00:06:36.805
times the limit

00:06:39.140 --> 00:06:40.807
as N approaches zero

00:06:41.653 --> 00:06:43.153
of the natural log

00:06:44.245 --> 00:06:45.078
of

00:06:46.820 --> 00:06:47.653
one

00:06:48.930 --> 00:06:49.847
one plus N.

00:06:51.287 --> 00:06:53.537
We do that in orange color.

00:06:54.700 --> 00:06:57.117
One plus N to the one over N.

00:06:58.492 --> 00:07:01.020
And now, what really gets affected

00:07:01.020 --> 00:07:02.846
is what's going on inside
of the natural log.

00:07:02.846 --> 00:07:04.434
That's where all of the Ns are.

00:07:04.434 --> 00:07:06.323
And so let's bring the
limit inside of that.

00:07:06.323 --> 00:07:08.447
So this is all going to be equal to

00:07:08.447 --> 00:07:09.992
give myself some space.

00:07:09.992 --> 00:07:11.930
A little bit of extra chalk board space.

00:07:11.930 --> 00:07:14.863
This is going to be equal to

00:07:14.863 --> 00:07:15.696
one over X

00:07:16.652 --> 00:07:18.760
times the natural log

00:07:18.760 --> 00:07:21.095
times the natural log of

00:07:21.095 --> 00:07:21.928
the limit

00:07:23.539 --> 00:07:25.206
as N approaches zero

00:07:26.608 --> 00:07:27.441
of

00:07:28.876 --> 00:07:29.709
one plus N

00:07:30.755 --> 00:07:32.255
to the one over N.

00:07:33.117 --> 00:07:35.029
Close those parenthesis.

00:07:35.029 --> 00:07:36.574
Now this is exciting.

00:07:36.574 --> 00:07:39.444
What is inside the natural log here?

00:07:39.444 --> 00:07:42.030
Well, this business
right over here, this is

00:07:42.030 --> 00:07:44.447
a definition of the number E.

00:07:45.335 --> 00:07:47.347
So that is equal to E.

00:07:47.347 --> 00:07:49.265
Well, what's the natural log of E?

00:07:49.265 --> 00:07:50.633
Well that's just one.

00:07:50.633 --> 00:07:53.004
So it's one over X times one.

00:07:53.004 --> 00:07:55.276
Well, that is indeed equal to

00:07:55.276 --> 00:07:56.496
one over X.

00:07:56.496 --> 00:07:58.070
Which is exactly

00:07:58.070 --> 00:08:00.084
the result that we were looking for.

00:08:00.084 --> 00:08:01.473
That the derivative

00:08:01.473 --> 00:08:03.370
with respect to X of natural log of X,

00:08:03.370 --> 00:08:04.537
is one over X.

00:08:06.417 --> 00:08:07.584
Very exciting.


Proof: The derivative of __ is __

WEBVTT
Kind: captions
Language: en

00:00:00.735 --> 00:00:02.468
- [Lecturer] The number e has all sorts

00:00:02.468 --> 00:00:04.080
of amazing properties.

00:00:04.080 --> 00:00:07.046
Just as a review you can
define it in terms of a limit.

00:00:07.046 --> 00:00:09.819
The limit as n approaches infinity of

00:00:09.819 --> 00:00:13.236
1 + 1/n to the nth power.

00:00:14.114 --> 00:00:19.064
You can also define it as
the limit as n approaches 0

00:00:19.064 --> 00:00:22.731
of 1 + n to the 1/nth power.

00:00:23.986 --> 00:00:25.645
What we're going to focus on this video

00:00:25.645 --> 00:00:27.299
is an amazing property of e,

00:00:27.299 --> 00:00:29.241
and e has many many amazing properties,

00:00:29.241 --> 00:00:31.017
but this is the one that's maybe

00:00:31.017 --> 00:00:32.134
the most relevant to calculus,

00:00:32.134 --> 00:00:33.589
and that's the notion that if I take

00:00:33.589 --> 00:00:38.428
the derivative with
respect to x of e to the x,

00:00:38.428 --> 00:00:41.327
then it is equal to, drumroll,

00:00:41.327 --> 00:00:43.564
equal to e to the x.

00:00:43.564 --> 00:00:45.349
That to me is amazing.

00:00:45.349 --> 00:00:46.963
Let's just appreciate it for a second

00:00:46.963 --> 00:00:48.870
before we actually prove it.

00:00:48.870 --> 00:00:52.126
This is part of the
graph of y = e to the x.

00:00:52.126 --> 00:00:55.277
And so what this says is,
the derivative of e to the x

00:00:55.277 --> 00:00:57.432
for any x is equal to e to the x!

00:00:57.432 --> 00:00:58.754
The slope of the tangent line

00:00:58.754 --> 00:01:01.814
at any point here is equal
to the value of the function.

00:01:01.814 --> 00:01:04.084
Let's just appreciate that.

00:01:04.084 --> 00:01:05.864
So right over here the
value of the function is 1,

00:01:05.864 --> 00:01:09.454
and the slope of the tangent line is 1.

00:01:09.454 --> 00:01:11.250
Here the value of the function is 2,

00:01:11.250 --> 00:01:14.233
and the slope of the tangent line is 2.

00:01:14.233 --> 00:01:16.858
Here the value of the function is 4,

00:01:16.858 --> 00:01:19.673
and the slope of the
tangent line is equal to 4.

00:01:19.673 --> 00:01:21.056
So I could just go on,

00:01:21.056 --> 00:01:24.540
this is just another
amazing thing about e,

00:01:24.540 --> 00:01:26.657
and we'll see many more in calculus,

00:01:26.657 --> 00:01:30.299
but let's now prove that
this is actually true.

00:01:30.299 --> 00:01:33.337
So let's just use our
definition of a derivative.

00:01:33.337 --> 00:01:37.504
So the derivative with
respect to x, of e to the x,

00:01:38.714 --> 00:01:42.881
would be the limit of delta
x, or as delta approaches 0,

00:01:44.768 --> 00:01:48.851
of e to the x + delta x, - e to the x,

00:01:50.842 --> 00:01:54.816
all of that over, all
of that over delta x.

00:01:54.816 --> 00:01:56.926
Now let's do some
algebraic manipulation here

00:01:56.926 --> 00:01:59.328
to see if we can make some sense of it.

00:01:59.328 --> 00:02:03.495
So this is going to be equal
to the limit as delta x

00:02:04.709 --> 00:02:07.238
approaches 0 of, let's see,

00:02:07.238 --> 00:02:09.927
what happens, well I
won't skip any steps here.

00:02:09.927 --> 00:02:14.473
This is the same thing as e to
the x times e to the delta x,

00:02:14.473 --> 00:02:16.386
this is just using our
exponent properties here

00:02:16.386 --> 00:02:19.136
minus e to the x over delta x.

00:02:21.101 --> 00:02:25.146
Just to be clear, I rewrote
this right over here

00:02:25.146 --> 00:02:27.146
as this right over here.

00:02:28.391 --> 00:02:31.982
Now I can factor out an e to the x,

00:02:31.982 --> 00:02:34.880
and in fact, because e
to the x is not affected

00:02:34.880 --> 00:02:37.779
as delta x approaches 0,

00:02:37.779 --> 00:02:41.142
I can factor the e to the
x out of the entire limit.

00:02:41.142 --> 00:02:43.798
So let's do that, let's take e to the x

00:02:43.798 --> 00:02:47.965
out of the entire limit, let's
factor it completely out.

00:02:49.127 --> 00:02:51.076
It does not get affected by the delta x.

00:02:51.076 --> 00:02:54.013
So this is going to be equal to,

00:02:54.013 --> 00:02:56.513
factor that e to the x out,

00:02:57.411 --> 00:03:01.578
e to the x times the limit
as delta x approaches 0,

00:03:03.571 --> 00:03:07.738
of e to the delta x - 1,
all of that over delta x.

00:03:12.037 --> 00:03:15.276
So now we're gonna get a little
bit fancy with our limits.

00:03:15.276 --> 00:03:18.381
We're gonna do what's known
as a change of variable.

00:03:18.381 --> 00:03:21.059
So I'm gonna say, well
let's see I don't know

00:03:21.059 --> 00:03:22.924
how to directly find this
limit right over here,

00:03:22.924 --> 00:03:24.500
but maybe I can simplify it,

00:03:24.500 --> 00:03:26.573
and who knows, maybe I can get it into

00:03:26.573 --> 00:03:28.994
one of these forms up here.

00:03:28.994 --> 00:03:30.985
So what if I were to
make the substitution,

00:03:30.985 --> 00:03:33.714
and let me do it over here.

00:03:33.714 --> 00:03:35.870
Let's say I would make
the substitution that

00:03:35.870 --> 00:03:38.620
n = e to the delta x - 1.

00:03:41.145 --> 00:03:44.381
So what would this be if we
were to solve for delta x?

00:03:44.381 --> 00:03:46.379
Well let's see, we could
add 1 to both sides,

00:03:46.379 --> 00:03:49.688
n + 1 = e to the delta x.

00:03:49.688 --> 00:03:51.746
To solve for delta x we could just take

00:03:51.746 --> 00:03:54.546
the natural logarithm,
log base e of both sides.

00:03:54.546 --> 00:03:58.713
And we would get the natural
log of n + 1 = delta x.

00:04:01.740 --> 00:04:04.924
And so, we can make that substitution,

00:04:04.924 --> 00:04:08.021
this could be replaced with this,

00:04:08.021 --> 00:04:10.124
and what we have in
the numerator over here

00:04:10.124 --> 00:04:13.541
can be replaced with n, right over there,

00:04:14.500 --> 00:04:16.473
and what would happen to the limit?

00:04:16.473 --> 00:04:19.223
Well, as delta x approaches 0,

00:04:20.554 --> 00:04:22.304
what does n approach?

00:04:23.258 --> 00:04:27.425
So delta x approaches 0
implies that n approaches what?

00:04:28.666 --> 00:04:30.322
Let's see, as delta x approaches 0,

00:04:30.322 --> 00:04:33.053
this would be e to the 0 which is 1 - 1,

00:04:33.053 --> 00:04:34.840
so looks like n approaches 0.

00:04:34.840 --> 00:04:36.218
And you can look at it over here,

00:04:36.218 --> 00:04:38.697
as n approaches 0, right over here,

00:04:38.697 --> 00:04:40.931
natural log of 0 + 1, natural log of 1,

00:04:40.931 --> 00:04:42.370
that is 0.

00:04:42.370 --> 00:04:44.798
So as each of them, as
delta x approaches 0,

00:04:44.798 --> 00:04:46.423
n approaches 0, as n approaches 0,

00:04:46.423 --> 00:04:47.667
delta x approaches 0.

00:04:47.667 --> 00:04:50.197
So then you can replace, if
you make the change of variable

00:04:50.197 --> 00:04:52.220
from delta x to n, you could still say

00:04:52.220 --> 00:04:54.363
as n approaches 0.

00:04:54.363 --> 00:04:56.403
So let me rewrite all of this.

00:04:56.403 --> 00:04:59.297
That was the fanciest
step in this entire thing

00:04:59.297 --> 00:05:01.002
that we're about to do.

00:05:01.002 --> 00:05:03.481
So this is gonna be e to the x times

00:05:03.481 --> 00:05:06.576
the limit, since we changed our variables

00:05:06.576 --> 00:05:09.107
now gonna be as n approaches 0,

00:05:09.107 --> 00:05:11.284
because as delta x
approaches 0, n approaches 0,

00:05:11.284 --> 00:05:12.526
and vice versa.

00:05:12.526 --> 00:05:14.058
And this numerator here, we said hey,

00:05:14.058 --> 00:05:16.725
that's gonna be equal to n over,

00:05:17.738 --> 00:05:21.905
delta x is now the natural
log of n + 1, ln(n+1).

00:05:24.421 --> 00:05:26.496
Now what does that do for us?

00:05:26.496 --> 00:05:29.472
Well what if we were
to divide the numerator

00:05:29.472 --> 00:05:31.750
and the denominator by n?

00:05:31.750 --> 00:05:34.272
So let's multiply down here by 1/n,

00:05:34.272 --> 00:05:38.243
and let's multiply up here by 1/n.

00:05:38.243 --> 00:05:40.356
Well our numerator's
just gonna be equal to 1.

00:05:40.356 --> 00:05:42.577
What does our denominator equal?

00:05:42.577 --> 00:05:44.684
Well here we can just use
our exponent properties.

00:05:44.684 --> 00:05:47.279
All of this is gonna be equal to,

00:05:47.279 --> 00:05:51.871
this is gonna be equal to, we
have our e to the x out front,

00:05:51.871 --> 00:05:55.121
e to the x, and then we have the limit,

00:05:58.652 --> 00:06:02.657
as n approaches 0, our numerator is now 1,

00:06:02.657 --> 00:06:05.740
over, and now I'm gonna rewrite this,

00:06:06.578 --> 00:06:09.507
just using our logarithm properties.

00:06:09.507 --> 00:06:11.620
If I have, I'll do it right over here.

00:06:11.620 --> 00:06:14.562
If I have a times the natural log of b,

00:06:14.562 --> 00:06:15.844
this is the same thing as

00:06:15.844 --> 00:06:20.495
the natural log of b to
the a power, the ath power,

00:06:20.495 --> 00:06:22.191
this is just natural log properties.

00:06:22.191 --> 00:06:24.051
So we have down here, this
would be the same thing

00:06:24.051 --> 00:06:28.218
as the natural log of n+1,
actually let me write it

00:06:29.150 --> 00:06:33.317
the other way around, 1+n,
1+n, I just swapped these two,

00:06:34.280 --> 00:06:39.096
to the 1/n power, so the
natural log of that whole thing.

00:06:39.096 --> 00:06:41.373
Now, you might be getting
that tingly feeling,

00:06:41.373 --> 00:06:44.226
because something is
starting to look familiar.

00:06:44.226 --> 00:06:47.039
What I just constructed
inside the logarithm here,

00:06:47.039 --> 00:06:50.764
looks an awful lot like what
we have right over here.

00:06:50.764 --> 00:06:52.381
And the limit as n approaches 0,

00:06:52.381 --> 00:06:54.695
limit as n approaches 0.

00:06:54.695 --> 00:06:57.190
And in fact we can use
our limit properties,

00:06:57.190 --> 00:06:59.228
this one isn't affected,
what's really affected

00:06:59.228 --> 00:07:01.718
is what's inside the logarithm.

00:07:01.718 --> 00:07:04.370
So we could say that this
is going to be equal to,

00:07:04.370 --> 00:07:06.145
and we're approaching our drumroll,

00:07:06.145 --> 00:07:10.312
e to the x times 1 over,
1 over the natural log,

00:07:14.777 --> 00:07:18.110
I'll do that blue color, natural log of,

00:07:19.357 --> 00:07:21.097
I'll give myself some space,

00:07:21.097 --> 00:07:24.264
the limit, the limit as n approaches 0

00:07:27.777 --> 00:07:30.527
of this business right over here,

00:07:32.074 --> 00:07:36.241
which I could just write
as, 1+n to the 1/nth power.

00:07:38.615 --> 00:07:40.268
So this is really interesting.

00:07:40.268 --> 00:07:42.172
What is this?

00:07:42.172 --> 00:07:44.340
What did I just, what do I
have there in the denominator

00:07:44.340 --> 00:07:45.332
with this limit?

00:07:45.332 --> 00:07:47.399
What is this thing equal to?

00:07:47.399 --> 00:07:50.537
Well we already said,
that is a definition of e.

00:07:50.537 --> 00:07:52.809
That is this right over here,

00:07:52.809 --> 00:07:55.036
which is equal to this over here.

00:07:55.036 --> 00:07:56.869
So this is equal to e.

00:07:57.816 --> 00:08:00.051
So what does this all boil down to?

00:08:00.051 --> 00:08:01.250
I think you see where this is going,

00:08:01.250 --> 00:08:03.730
but this is fun, we're downhill from here.

00:08:03.730 --> 00:08:07.897
This is e to the x times 1/ln(e).

00:08:13.475 --> 00:08:16.464
Well, ln(e), what power
do I have to raise e

00:08:16.464 --> 00:08:17.451
to get to e?

00:08:17.451 --> 00:08:19.026
Well I just have to raise it to 1.

00:08:19.026 --> 00:08:22.609
So this gets us e to
the x, and we're done.

00:08:24.344 --> 00:08:27.447
We've just proven that the
derivative with respect to x

00:08:27.447 --> 00:08:30.884
of e to the x is indeed
equal to e to the x.

00:08:30.884 --> 00:08:33.947
That's an amazing finding, that shows us

00:08:33.947 --> 00:08:38.030
one more dimension of the
beauty of the number e.


Proof of the derivative of cos(x)

WEBVTT
Kind: captions
Language: en

00:00:00.010 --> 00:00:01.576
- [Instructor] What I
wanna do in this video

00:00:01.576 --> 00:00:04.792
is make a visual argument as
to why the derivative with

00:00:04.792 --> 00:00:08.959
respect to x of cosine of x is
equal to negative sine of x.

00:00:11.955 --> 00:00:13.867
And we're gonna base this argument,

00:00:13.867 --> 00:00:16.366
based on a previous proof we made

00:00:16.366 --> 00:00:21.308
that the derivative with
respect to x of sine of x

00:00:21.308 --> 00:00:23.308
is equal to cosine of x.

00:00:25.630 --> 00:00:27.439
So we're gonna assume this over here.

00:00:27.439 --> 00:00:28.807
I encourage you to watch that video.

00:00:28.807 --> 00:00:31.431
That's actually a fairly
involved proof that proves this,

00:00:31.431 --> 00:00:34.059
but if we assume this, I'm
gonna make a visual argument

00:00:34.059 --> 00:00:36.892
that this right over
here is the derivative

00:00:36.892 --> 00:00:40.360
with respect to x of cosine
of x is negative sine of x.

00:00:40.360 --> 00:00:43.743
So right over here we
seen sine of x in red

00:00:43.743 --> 00:00:46.263
and we see cosine of x in blue.

00:00:46.263 --> 00:00:48.000
And we're assuming that this blue graph

00:00:48.000 --> 00:00:50.987
is showing the derivative,
the slope of the tangent line

00:00:50.987 --> 00:00:53.374
for any x value of the red graph.

00:00:53.374 --> 00:00:55.688
And we've got an intuition
for that in previous videos.

00:00:55.688 --> 00:00:57.234
Now what I'm gonna do next,

00:00:57.234 --> 00:00:59.937
is I'm gonna shift both of these graphs

00:00:59.937 --> 00:01:02.187
to the left by pi over two.

00:01:03.589 --> 00:01:06.506
Shift it to the left by pi over two

00:01:07.867 --> 00:01:11.202
and I'm also gonna shift
the blue graph to the left

00:01:11.202 --> 00:01:12.452
by pi over two.

00:01:14.876 --> 00:01:16.828
And so what am I going to get?

00:01:16.828 --> 00:01:19.089
Well the blue graph is
gonna look like this one

00:01:19.089 --> 00:01:22.300
right over here and if it
was cosine of x up here,

00:01:22.300 --> 00:01:25.002
we can now say that this is equal to

00:01:25.002 --> 00:01:28.835
y is equal to cosine
of x plus pi over two.

00:01:31.279 --> 00:01:32.225
This is the blue graph,

00:01:32.225 --> 00:01:34.978
cosine of x, shifted to
the left by pi over two.

00:01:34.978 --> 00:01:39.145
And this is y is equal to
sine of x plus pi over two.

00:01:44.116 --> 00:01:46.636
Now the visual argument is, all I did,

00:01:46.636 --> 00:01:48.814
is I shifted both of these graphs

00:01:48.814 --> 00:01:50.913
to the left by pi over two.

00:01:50.913 --> 00:01:52.968
So it should still be the
case that the derivative

00:01:52.968 --> 00:01:56.115
of the red graph is the blue graph.

00:01:56.115 --> 00:01:57.688
So we should still be able to say

00:01:57.688 --> 00:02:01.015
that the derivative with
respect to x of the red graph,

00:02:01.015 --> 00:02:03.348
sin of x plus pi over two

00:02:06.385 --> 00:02:09.879
that that is equal to the blue graph.

00:02:09.879 --> 00:02:14.046
That that is equal to cosine
of x plus pi over two.

00:02:16.595 --> 00:02:19.815
Now what is sin of x plus pi over two?

00:02:19.815 --> 00:02:23.916
Well that's the same thing as cosine of x.

00:02:23.916 --> 00:02:26.288
You can see this red graph is
the same thing as cosine of x.

00:02:26.288 --> 00:02:27.685
We know that from our trig identities

00:02:27.685 --> 00:02:30.210
and you can also see in
intuitively or graphically

00:02:30.210 --> 00:02:31.938
just by looking at these graphs.

00:02:31.938 --> 00:02:35.219
Now what is cosine of x plus pi over two?

00:02:35.219 --> 00:02:36.950
Well once again, from our trig identities,

00:02:36.950 --> 00:02:39.315
we know that that is the exact same thing

00:02:39.315 --> 00:02:41.148
as negative sine of x.

00:02:43.015 --> 00:02:44.955
So there you have it, the visual argument.

00:02:44.955 --> 00:02:46.722
Just start with this knowledge,

00:02:46.722 --> 00:02:50.156
shift both of these graphs
to the left by pi over two,

00:02:50.156 --> 00:02:51.649
it should still be true,

00:02:51.649 --> 00:02:53.488
that the derivative with respect to x

00:02:53.488 --> 00:02:56.155
of sine of x plus pi over two

00:02:58.069 --> 00:03:01.736
is equal to cosine of x plus pi over two.

00:03:05.237 --> 00:03:07.070
And this is the same thing
as saying what we have

00:03:07.070 --> 00:03:08.431
right over here.

00:03:08.431 --> 00:03:10.124
So now we should feel pretty good.

00:03:10.124 --> 00:03:11.700
We proved this in a previous video

00:03:11.700 --> 00:03:14.522
and we have a very strong
visual argument for this

00:03:14.522 --> 00:03:16.924
for cosine of x in this video.


Derivatives of sin(x) and cos(x)

WEBVTT
Kind: captions
Language: en

00:00:00.209 --> 00:00:02.230
- [Instructor] What I'd
like to do in this video

00:00:02.230 --> 00:00:04.424
is get an intuitive sense
for what the derivative

00:00:04.424 --> 00:00:07.034
with respect to x of sine of x is

00:00:07.034 --> 00:00:10.575
and what the derivative with
respect to x of cosine of x is.

00:00:10.575 --> 00:00:13.491
And I've graphed y is equal
to cosine of x in blue

00:00:13.491 --> 00:00:16.765
and y is equal to sine of x in red.

00:00:16.765 --> 00:00:18.585
We're not going to prove
what the derivatives are,

00:00:18.585 --> 00:00:21.131
but we're gonna know what they
are, get an intuitive sense

00:00:21.131 --> 00:00:23.600
and in future videos
we'll actually do a proof.

00:00:23.600 --> 00:00:25.333
So let's start with sine of x.

00:00:25.333 --> 00:00:26.746
So the derivative can be viewed

00:00:26.746 --> 00:00:28.486
as the slope of the tangent line.

00:00:28.486 --> 00:00:31.185
So for example at this
point right over here,

00:00:31.185 --> 00:00:35.352
it looks like the slope of our
tangent line should be zero.

00:00:36.224 --> 00:00:40.391
So our derivative function
should be zero at that x value.

00:00:41.875 --> 00:00:46.042
Similarly, over here, it looks
like the derivative is zero.

00:00:46.940 --> 00:00:49.642
Slope of the tangent line would be zero.

00:00:49.642 --> 00:00:53.441
So whatever our derivative
function is at that x value,

00:00:53.441 --> 00:00:55.691
it should be equal to zero.

00:00:57.233 --> 00:00:59.333
If we look right over here on sine of x,

00:00:59.333 --> 00:01:02.389
it looks like the slope
of the tangent line

00:01:02.389 --> 00:01:05.269
would be pretty close to one.

00:01:05.269 --> 00:01:08.110
If that is the case, then
in our derivative function

00:01:08.110 --> 00:01:09.573
when x is equal to zero

00:01:09.573 --> 00:01:12.692
that derivative function
should be equal to one.

00:01:12.692 --> 00:01:14.914
Similarly, over here, it looks like

00:01:14.914 --> 00:01:17.953
the slope of the tangent
line is negative one,

00:01:17.953 --> 00:01:20.494
which tells us that
the derivative function

00:01:20.494 --> 00:01:24.661
should be hitting the value of
negative one at that x value.

00:01:25.502 --> 00:01:28.590
So you're probably seeing
something interesting emerge.

00:01:28.590 --> 00:01:31.521
Everywhere, while we're trying to plot

00:01:31.521 --> 00:01:34.218
the slope of the tangent
line, it seems to coincide

00:01:34.218 --> 00:01:37.368
with y is equal to cosine of x.

00:01:37.368 --> 00:01:40.700
And it is indeed the case that
the derivative of sine of x

00:01:40.700 --> 00:01:43.278
is equal to cosine of x.

00:01:43.278 --> 00:01:44.717
And you can see that it makes sense,

00:01:44.717 --> 00:01:47.058
not just at the points we
tried, but even in the trends.

00:01:47.058 --> 00:01:50.097
If you look at sine of x
here, the slope is one,

00:01:50.097 --> 00:01:52.859
but then it becomes less
and less and less positive

00:01:52.859 --> 00:01:54.416
all the way until it becomes zero.

00:01:54.416 --> 00:01:57.446
Cosine of x, the value
of the function is one

00:01:57.446 --> 00:01:59.298
and it becomes less and less positive

00:01:59.298 --> 00:02:01.733
all the way until it equals zero.

00:02:01.733 --> 00:02:03.684
And you could keep doing
that type of analysis

00:02:03.684 --> 00:02:05.006
to feel good about it.

00:02:05.006 --> 00:02:08.370
In another video we're going
to prove this more rigorously.

00:02:08.370 --> 00:02:10.580
So now let's think about cosine of x.

00:02:10.580 --> 00:02:13.703
So cosine of x, right over here,

00:02:13.703 --> 00:02:15.712
the slope of the tangent line

00:02:15.712 --> 00:02:17.786
looks like it is zero.

00:02:17.786 --> 00:02:19.109
And so it's derivative function

00:02:19.109 --> 00:02:21.090
needs to be zero at that point.

00:02:21.090 --> 00:02:23.229
So, hey, maybe it's sine of x.

00:02:23.229 --> 00:02:24.867
Let's keep trying this.

00:02:24.867 --> 00:02:27.747
So over here, cosine of x,

00:02:27.747 --> 00:02:32.727
it looks like the slope of the
tangent line is negative one

00:02:32.727 --> 00:02:36.240
and so we would want the
derivative to go through

00:02:36.240 --> 00:02:38.695
that point right over there.

00:02:38.695 --> 00:02:40.337
All right this is starting to seem,

00:02:40.337 --> 00:02:41.889
it doesn't seem like the
derivative of cosine of x

00:02:41.889 --> 00:02:43.335
could be sine of x.

00:02:43.335 --> 00:02:46.040
In fact, this is the opposite
of what sine of x is doing.

00:02:46.040 --> 00:02:49.008
Sine of x is at one, not
negative one at that point.

00:02:49.008 --> 00:02:50.266
But that's an interesting theory,

00:02:50.266 --> 00:02:54.712
maybe the derivative of cosine
of x is negative sine of x.

00:02:54.712 --> 00:02:56.295
So let's plot that.

00:02:59.531 --> 00:03:01.388
So this does seem to coincide.

00:03:01.388 --> 00:03:04.779
The derivative of cosine of x
here looks like negative one,

00:03:04.779 --> 00:03:06.102
the slope of a tangent line

00:03:06.102 --> 00:03:10.026
and negative sign of this
x value is negative one.

00:03:10.026 --> 00:03:12.354
Over here the derivative of cosine of x

00:03:12.354 --> 00:03:14.061
looks like it is zero

00:03:14.061 --> 00:03:17.962
and negative sine of x is indeed zero.

00:03:17.962 --> 00:03:20.364
So it actually turns
out that it is the case,

00:03:20.364 --> 00:03:24.531
that the derivative of cosine
of x is negative sine of x.

00:03:25.822 --> 00:03:27.471
So these are really good to know.

00:03:27.471 --> 00:03:29.061
These are kind of fundamental

00:03:29.061 --> 00:03:31.194
trigonometric derivatives to know.

00:03:31.194 --> 00:03:33.238
We'll be able to derive
other things for them.

00:03:33.238 --> 00:03:35.690
And hopefully this video gives
you a good intuitive sense

00:03:35.690 --> 00:03:37.072
of why this is true.

00:03:37.072 --> 00:03:40.655
And in future videos, we
will prove it rigorously.


Limit of sin(x)/x as x approaches 0

WEBVTT
Kind: captions
Language: en

00:00:00.251 --> 00:00:02.158
- [Instructor] What we're
going to do in this video

00:00:02.158 --> 00:00:04.883
is prove that the limit

00:00:04.883 --> 00:00:07.358
as theta approaches zero

00:00:07.358 --> 00:00:08.691
of sine of theta

00:00:10.196 --> 00:00:13.082
over theta is equal to one.

00:00:13.082 --> 00:00:15.599
So let's start with a
little bit of a geometric

00:00:15.599 --> 00:00:19.149
or trigonometric construction
that I have here.

00:00:19.149 --> 00:00:22.675
So this white circle,
this is a unit circle,

00:00:22.675 --> 00:00:24.251
that we'll label it as such.

00:00:24.251 --> 00:00:26.091
So it has radius one,

00:00:26.091 --> 00:00:27.607
unit

00:00:27.607 --> 00:00:28.440
circle.

00:00:29.432 --> 00:00:31.183
So what does the length

00:00:31.183 --> 00:00:34.907
of this salmon-colored line represent?

00:00:34.907 --> 00:00:38.482
Well, the height of this line
would be the y-coordinate

00:00:38.482 --> 00:00:42.407
of where this radius
intersects the unit circle.

00:00:42.407 --> 00:00:45.242
And so by definition, by
the unit circle definition

00:00:45.242 --> 00:00:48.425
of trig functions, the length of this line

00:00:48.425 --> 00:00:50.842
is going to be sine of theta.

00:00:52.307 --> 00:00:55.075
If we wanted to make sure
that also worked for thetas

00:00:55.075 --> 00:00:58.000
that end up in the fourth
quadrant, which will be useful,

00:00:58.000 --> 00:01:00.983
we can just insure that
it's the absolute value

00:01:00.983 --> 00:01:02.825
of the sine of theta.

00:01:02.825 --> 00:01:05.292
Now what about this blue line over here?

00:01:05.292 --> 00:01:08.992
Can I express that in terms
of a trigonometric function?

00:01:08.992 --> 00:01:09.825
Well, let's think about it.

00:01:09.825 --> 00:01:12.050
What would tangent of theta be?

00:01:12.050 --> 00:01:13.474
Let me write it over here.

00:01:13.474 --> 00:01:15.632
Tangent of theta

00:01:15.632 --> 00:01:18.549
is equal to opposite over adjacent.

00:01:21.190 --> 00:01:24.242
So if we look at this broader
triangle right over here,

00:01:24.242 --> 00:01:27.165
this is our angle theta in radians.

00:01:27.165 --> 00:01:29.484
This is the opposite side.

00:01:29.484 --> 00:01:32.559
The adjacent side down here,
this just has length one.

00:01:32.559 --> 00:01:34.624
Remember, this is a unit circle.

00:01:34.624 --> 00:01:36.424
So this just has length one,

00:01:36.424 --> 00:01:39.716
so the tangent of theta
is the opposite side.

00:01:39.716 --> 00:01:43.971
The opposite side is equal
to the tangent of theta.

00:01:43.971 --> 00:01:46.484
And just like before, this is
going to be a positive value

00:01:46.484 --> 00:01:48.475
for sitting here in the first quadrant

00:01:48.475 --> 00:01:49.690
but I want things to work

00:01:49.690 --> 00:01:51.958
in both the first and the fourth quadrant

00:01:51.958 --> 00:01:53.357
for the sake of our proof,

00:01:53.357 --> 00:01:56.133
so I'm just gonna put
an absolute value here.

00:01:56.133 --> 00:01:57.432
So now that we've done,

00:01:57.432 --> 00:01:59.324
I'm gonna think about some triangles

00:01:59.324 --> 00:02:01.307
and their respective areas.

00:02:01.307 --> 00:02:03.400
So first, I'm gonna draw a triangle

00:02:03.400 --> 00:02:07.225
that sits in this wedge,
in this pie piece,

00:02:07.225 --> 00:02:10.324
this pie slice within the circle,

00:02:10.324 --> 00:02:13.074
so I can construct this triangle.

00:02:14.382 --> 00:02:16.234
And so let's think about the area

00:02:16.234 --> 00:02:20.217
of what I am shading in right over here.

00:02:20.217 --> 00:02:22.550
How can I express that area?

00:02:23.582 --> 00:02:24.774
Well, it's a triangle.

00:02:24.774 --> 00:02:26.349
We know that the area of a triangle

00:02:26.349 --> 00:02:29.750
is 1/2 base times height.

00:02:29.750 --> 00:02:30.741
We know the height

00:02:30.741 --> 00:02:32.792
is the absolute value of the sine of theta

00:02:32.792 --> 00:02:35.758
and we know that the base is equal to one,

00:02:35.758 --> 00:02:39.925
so the area here is
going to be equal to 1/2

00:02:41.258 --> 00:02:43.824
times our base, which is one,

00:02:43.824 --> 00:02:44.842
times our height,

00:02:44.842 --> 00:02:48.192
which is the absolute
value of the sine of theta.

00:02:48.192 --> 00:02:49.433
I'll rewrite it over here.

00:02:49.433 --> 00:02:50.341
I can just rewrite that

00:02:50.341 --> 00:02:54.791
as the absolute value of
the sine of theta over two.

00:02:54.791 --> 00:02:58.458
Now let's think about
the area of this wedge

00:02:58.458 --> 00:03:02.608
that I am highlighting
in this yellow color.

00:03:02.608 --> 00:03:07.075
So what fraction of the entire
circle is this going to be?

00:03:07.075 --> 00:03:08.833
If I were to go all the
way around the circle,

00:03:08.833 --> 00:03:10.335
it would be two pi radians,

00:03:10.335 --> 00:03:14.502
so this is theta over to
two pis of the entire circle

00:03:15.607 --> 00:03:17.167
and we know the area of the circle.

00:03:17.167 --> 00:03:19.724
This is a unit circle,
it has a radius one,

00:03:19.724 --> 00:03:21.399
so it'd be times the area of the circle,

00:03:21.399 --> 00:03:24.017
which would be pi times the radius square,

00:03:24.017 --> 00:03:26.666
the radius is one, so it's
just gonna be times pi.

00:03:26.666 --> 00:03:29.342
And so the area of this
wedge right over here,

00:03:29.342 --> 00:03:30.592
theta over two.

00:03:31.466 --> 00:03:33.200
And if we wanted to make this work

00:03:33.200 --> 00:03:35.689
for thetas in the fourth quadrant,

00:03:35.689 --> 00:03:38.900
we could just write an absolute
value sign right over there

00:03:38.900 --> 00:03:42.082
'cause we're talking about positive area.

00:03:42.082 --> 00:03:44.999
And now let's think about
this larger triangle

00:03:44.999 --> 00:03:48.807
in this blue color, and this
is pretty straightforward.

00:03:48.807 --> 00:03:53.149
The area here is gonna be
1/2 times base times height.

00:03:53.149 --> 00:03:57.333
So the area, and once again,
this is this entire are,

00:03:57.333 --> 00:04:01.413
that's going to be 1/2 times
our base, which is one,

00:04:01.413 --> 00:04:02.246
times our height,

00:04:02.246 --> 00:04:05.808
which is the absolute
value of tangent of theta.

00:04:05.808 --> 00:04:07.158
And so I can just write that down

00:04:07.158 --> 00:04:11.325
as the absolute value of the
tangent of theta over two.

00:04:12.439 --> 00:04:14.550
Now, how would you compare the areas

00:04:14.550 --> 00:04:18.532
of this pink or this
salmon-colored triangle

00:04:18.532 --> 00:04:20.614
which sits inside of this wedge

00:04:20.614 --> 00:04:21.991
and how do you compare
that area of the wedge

00:04:21.991 --> 00:04:23.641
to the bigger triangle?

00:04:23.641 --> 00:04:25.899
Well, it's clear that the
area of the salmon triangle

00:04:25.899 --> 00:04:28.341
is less than or equal
to the area of the wedge

00:04:28.341 --> 00:04:30.782
and the area of the wedge
is less than or equal to

00:04:30.782 --> 00:04:33.415
the area of the big, blue triangle.

00:04:33.415 --> 00:04:35.492
The wedge includes the salmon triangle

00:04:35.492 --> 00:04:37.633
plus this area right over here,

00:04:37.633 --> 00:04:40.766
and then the blue triangle
includes the wedge

00:04:40.766 --> 00:04:43.993
plus it has this area right over here.

00:04:43.993 --> 00:04:45.592
So I think we can feel good visually

00:04:45.592 --> 00:04:48.058
that this statement
right over here is true

00:04:48.058 --> 00:04:48.891
and I'm just gonna do

00:04:48.891 --> 00:04:50.900
a little bit of algebraic manipulation.

00:04:50.900 --> 00:04:53.450
Let me multiply everything by two

00:04:53.450 --> 00:04:56.790
so I can rewrite that the absolute value

00:04:56.790 --> 00:05:00.514
of sine of theta is less than or equal to

00:05:00.514 --> 00:05:03.124
the absolute value of theta

00:05:03.124 --> 00:05:07.341
which is less than or
equal to the absolute value

00:05:07.341 --> 00:05:10.149
of tangent of theta, and let's see.

00:05:10.149 --> 00:05:11.699
Actually, instead of
writing the absolute value

00:05:11.699 --> 00:05:13.766
of tangent of theta,
I'm gonna rewrite that

00:05:13.766 --> 00:05:16.933
as the absolute value of sine of theta

00:05:18.359 --> 00:05:22.399
over the absolute value
of cosine of theta.

00:05:22.399 --> 00:05:23.524
That's gonna be the same thing

00:05:23.524 --> 00:05:27.007
as the absolute value of tangent of theta.

00:05:27.007 --> 00:05:27.957
And the reason why I did that

00:05:27.957 --> 00:05:29.775
is we can now divide everything

00:05:29.775 --> 00:05:32.200
by the absolute value of sine of theta.

00:05:32.200 --> 00:05:34.391
Since we're dividing
by a positive quantity,

00:05:34.391 --> 00:05:37.384
it's not going to change the
direction of the inequalities.

00:05:37.384 --> 00:05:38.983
So let's do that

00:05:38.983 --> 00:05:40.767
I'm gonna divide this

00:05:40.767 --> 00:05:44.270
by an absolute value of sine of theta.

00:05:44.270 --> 00:05:45.550
I'm gonna divide this

00:05:45.550 --> 00:05:49.091
by an absolute value of the sine of theta

00:05:49.091 --> 00:05:50.499
and then I'm gonna divide this

00:05:50.499 --> 00:05:54.083
by an absolute value of the sine of theta.

00:05:54.083 --> 00:05:55.654
And what do I get?

00:05:55.654 --> 00:05:57.987
Well, over here, I get a one

00:05:59.168 --> 00:06:00.807
and on the right-hand side, I get a one

00:06:00.807 --> 00:06:03.456
over the absolute value of cosine theta.

00:06:03.456 --> 00:06:05.206
These two cancel out.

00:06:06.466 --> 00:06:08.216
So the next step I'm gonna do

00:06:08.216 --> 00:06:11.142
is take the reciprocal of everything.

00:06:11.142 --> 00:06:13.141
And so when I take the
reciprocal of everything,

00:06:13.141 --> 00:06:15.932
that actually will
switch the inequalities.

00:06:15.932 --> 00:06:19.034
The reciprocal of one
is still going to be one

00:06:19.034 --> 00:06:20.983
but now, since I'm taking
the reciprocal of this here,

00:06:20.983 --> 00:06:24.340
it's gonna be greater than or equal to

00:06:24.340 --> 00:06:27.590
the absolute value of the sine of theta

00:06:28.474 --> 00:06:31.025
over the absolute value of theta,

00:06:31.025 --> 00:06:35.500
and that's going to be
greater than or equal to

00:06:35.500 --> 00:06:36.474
the reciprocal of one

00:06:36.474 --> 00:06:39.293
over the absolute value of cosine of theta

00:06:39.293 --> 00:06:42.710
is the absolute value of cosine of theta.

00:06:43.650 --> 00:06:46.574
We really just care about the
first and fourth quadrants.

00:06:46.574 --> 00:06:48.383
You can think about this theta

00:06:48.383 --> 00:06:50.332
approaching zero from that direction

00:06:50.332 --> 00:06:52.557
or from that direction there,

00:06:52.557 --> 00:06:55.242
so that would be the first
and fourth quadrants.

00:06:55.242 --> 00:06:58.733
So if we're in the first
quadrant and theta is positive,

00:06:58.733 --> 00:07:01.108
sine of theta is gonna
be positive as well.

00:07:01.108 --> 00:07:03.529
And if we're in the fourth
quadrant and theta's negative,

00:07:03.529 --> 00:07:05.084
well, sine of theta is
gonna have the same sign.

00:07:05.084 --> 00:07:07.125
It's going to be negative as well.

00:07:07.125 --> 00:07:09.358
And so these absolute value
signs aren't necessary.

00:07:09.358 --> 00:07:10.241
In the first quadrant,

00:07:10.241 --> 00:07:12.633
sine of theta and theta are both positive.

00:07:12.633 --> 00:07:14.592
In the fourth quadrant,
they're both negative,

00:07:14.592 --> 00:07:15.633
but when you divide them,

00:07:15.633 --> 00:07:20.299
you're going to get a positive
value, so I can erase those.

00:07:20.299 --> 00:07:22.108
If we're in the first or fourth quadrant,

00:07:22.108 --> 00:07:24.259
our X value is not negative,

00:07:24.259 --> 00:07:26.900
and so cosine of theta,
which is the x-coordinate

00:07:26.900 --> 00:07:29.858
on our unit circle, is
not going to be negative,

00:07:29.858 --> 00:07:33.250
and so we don't need the
absolute value signs over there.

00:07:33.250 --> 00:07:34.917
Now, we should pause a second

00:07:34.917 --> 00:07:37.642
because we're actually almost done.

00:07:37.642 --> 00:07:39.383
We have just set up three functions.

00:07:39.383 --> 00:07:42.492
You could think of this
as f of x is equal to,

00:07:42.492 --> 00:07:44.542
you could view this as f
of theta is equal to one,

00:07:44.542 --> 00:07:45.938
g of theta is equal to this,

00:07:45.938 --> 00:07:47.908
and h of theta is equal to that.

00:07:47.908 --> 00:07:49.882
And over the interval that we care about,

00:07:49.882 --> 00:07:52.965
we could say for negative pi over two

00:07:53.891 --> 00:07:57.132
is less than theta is
less than pi over two,

00:07:57.132 --> 00:07:59.599
but over this interval,
this is true for any theta

00:07:59.599 --> 00:08:01.925
over which these functions are defined.

00:08:01.925 --> 00:08:04.600
Sine of theta over theta is
defined over this interval,

00:08:04.600 --> 00:08:06.917
except where theta is equal to zero.

00:08:06.917 --> 00:08:08.599
But since we're defined everywhere else,

00:08:08.599 --> 00:08:11.316
we can now find the limit.

00:08:11.316 --> 00:08:14.716
So what we can say is,
well, by the squeeze theorem

00:08:14.716 --> 00:08:16.358
or by the sandwich theorem,

00:08:16.358 --> 00:08:18.283
if this is true over the interval,

00:08:18.283 --> 00:08:20.574
then we also know that
the following is true.

00:08:20.574 --> 00:08:23.300
And this, we deserve a
little bit of a drum roll.

00:08:23.300 --> 00:08:24.908
The limit

00:08:24.908 --> 00:08:27.575
as theta approaches zero of this

00:08:28.883 --> 00:08:33.474
is going to be greater
than or equal to the limit

00:08:33.474 --> 00:08:36.532
as theta approaches zero of this,

00:08:36.532 --> 00:08:37.891
which is the one that we care about,

00:08:37.891 --> 00:08:40.440
sine of theta over theta,

00:08:40.440 --> 00:08:45.375
which is going to be greater
than or equal to the limit

00:08:45.375 --> 00:08:48.125
as theta approaches zero of this.

00:08:49.717 --> 00:08:53.600
Now this is clearly going
to be just equal to one.

00:08:53.600 --> 00:08:55.842
This is what we care about.

00:08:55.842 --> 00:08:58.050
And this, what's the limit
as theta approaches zero

00:08:58.050 --> 00:08:59.457
of cosine of theta?

00:08:59.457 --> 00:09:01.158
Well, cosine of zero is just one

00:09:01.158 --> 00:09:02.799
and it's a continuous function,

00:09:02.799 --> 00:09:04.599
so this is just gonna be one.

00:09:04.599 --> 00:09:05.432
So let's see.

00:09:05.432 --> 00:09:07.392
This limit is going to be
less than or equal to one

00:09:07.392 --> 00:09:09.347
and it's gonna be greater
than or equal to one,

00:09:09.347 --> 00:09:11.680
so this must be equal to one

00:09:13.832 --> 00:09:14.929
and we are done.


Limit of (1-cos(x))/x as x approaches 0

WEBVTT
Kind: captions
Language: en

00:00:00.351 --> 00:00:01.650
- [Instructor] What we
wanna do in this video

00:00:01.650 --> 00:00:05.898
is figure out what the
limit, as x approaches zero,

00:00:05.898 --> 00:00:10.065
of one minus cosine of
x over x is equal to.

00:00:10.940 --> 00:00:13.662
And we're going to assume we
know one thing ahead of time.

00:00:13.662 --> 00:00:15.882
We're going to assume
we know that the limit,

00:00:15.882 --> 00:00:19.882
as x approaches zero, of sine of x over x,

00:00:21.807 --> 00:00:23.819
that this is equal to one.

00:00:23.819 --> 00:00:25.551
Now, I'm not gonna reprove
this in this video,

00:00:25.551 --> 00:00:26.861
but we have a whole other video

00:00:26.861 --> 00:00:30.511
dedicated to proving this famous limit,

00:00:30.511 --> 00:00:33.885
and we do it using the squeeze,
or the sandwich theorem.

00:00:33.885 --> 00:00:36.316
So, let's see if we can work this out.

00:00:36.316 --> 00:00:37.726
So the first thing we're going to do

00:00:37.726 --> 00:00:40.306
is algebraically
manipulate this expression.

00:00:40.306 --> 00:00:42.256
What I'm going to do is
I'm going to multiply

00:00:42.256 --> 00:00:44.171
both the numerator and the denominator

00:00:44.171 --> 00:00:46.171
by one plus cosine of x.

00:00:49.278 --> 00:00:50.242
So, times...

00:00:50.242 --> 00:00:51.630
The denominator I have
to do the same thing,

00:00:51.630 --> 00:00:53.531
one plus cosine of x.

00:00:53.531 --> 00:00:55.088
I'm not changing the
value of the expression,

00:00:55.088 --> 00:00:57.355
this is just multiplying it by one.

00:00:57.355 --> 00:00:59.019
What does that do for us?

00:00:59.019 --> 00:01:00.352
Well I can rewrite the whole thing as

00:01:00.352 --> 00:01:03.269
the limit, as x approaches zero,

00:01:04.521 --> 00:01:08.596
so one minus cosine of x
times one plus cosine of x,

00:01:08.596 --> 00:01:10.424
well that is just going to be...

00:01:10.424 --> 00:01:12.591
Put this in another color.

00:01:13.672 --> 00:01:18.339
That is going to be one
squared, which is just one,

00:01:18.339 --> 00:01:20.796
minus cosine squared of x.

00:01:20.796 --> 00:01:23.826
Cosine squared of x,
difference of squares.

00:01:23.826 --> 00:01:27.731
And then in the denominator,
I am going to have

00:01:27.731 --> 00:01:31.898
these, which is just x
times 1 plus cosine of x.

00:01:36.095 --> 00:01:39.185
Now what is one minus cosine squared of x?

00:01:39.185 --> 00:01:40.074
Well, this comes straight out of the

00:01:40.074 --> 00:01:42.890
Pythagorean identity, trig identity.

00:01:42.890 --> 00:01:46.981
This is the same thing
as the sine squared of x.

00:01:46.981 --> 00:01:48.814
So, sine squared of x.

00:01:50.220 --> 00:01:52.981
And so, I can rewrite all
of this as being equal to

00:01:52.981 --> 00:01:55.898
the limit, as x approaches zero,

00:01:57.971 --> 00:01:59.815
and let me rewrite this as...

00:01:59.815 --> 00:02:01.256
Instead of sine squared of x,

00:02:01.256 --> 00:02:03.882
that's the same thing as
sine of x times sine of x.

00:02:03.882 --> 00:02:04.753
Let me write it that way.

00:02:04.753 --> 00:02:06.670
Sine x times sine x.

00:02:09.246 --> 00:02:11.547
So, I'll take the first sine of x,

00:02:11.547 --> 00:02:14.450
so I'll take this one right over here,

00:02:14.450 --> 00:02:16.282
and put it over this x.

00:02:16.282 --> 00:02:18.199
So, sine of x over x

00:02:19.787 --> 00:02:22.870
times the second sine of x, this one,

00:02:24.073 --> 00:02:26.879
over one plus cosine of x

00:02:26.879 --> 00:02:30.629
times sine of x over one plus cosine of x.

00:02:32.765 --> 00:02:35.742
All I've done is I've leveraged
a trigonometric identity,

00:02:35.742 --> 00:02:37.872
and I've done a little bit
of algebraic manipulation.

00:02:37.872 --> 00:02:40.541
Well here, the limit of the product

00:02:40.541 --> 00:02:42.382
of these two expressions, is
going to be the same thing

00:02:42.382 --> 00:02:43.957
as the product of the limits.

00:02:43.957 --> 00:02:47.207
So I can rewrite this as being equal to

00:02:48.370 --> 00:02:52.537
the limit, as x approaches
zero, of sine of x over x

00:02:55.772 --> 00:02:59.189
times the limit, as x approaches zero,

00:03:01.935 --> 00:03:05.435
of sine of x over one plus cosine of x.

00:03:07.826 --> 00:03:09.661
Now, we said, going into this video,

00:03:09.661 --> 00:03:11.909
that we're going to assume
that we know what this is.

00:03:11.909 --> 00:03:13.571
We've proven it in other videos.

00:03:13.571 --> 00:03:15.339
What is the limit, as x approaches zero,

00:03:15.339 --> 00:03:17.054
of sine of x over x?

00:03:17.054 --> 00:03:19.781
Well, that is equal to one.

00:03:19.781 --> 00:03:21.891
So, this whole limit is
just going to be dependent

00:03:21.891 --> 00:03:23.607
on whatever this is equal to.

00:03:23.607 --> 00:03:26.215
Well, this is pretty
straight forward, here.

00:03:26.215 --> 00:03:29.333
As x approaches zero, the
numerator's approaching zero,

00:03:29.333 --> 00:03:30.793
sine of zero is zero.

00:03:30.793 --> 00:03:32.672
The denominator is approaching...

00:03:32.672 --> 00:03:34.095
Cosine of zero is one,

00:03:34.095 --> 00:03:36.133
so the denominator is approaching two.

00:03:36.133 --> 00:03:39.630
So this is approaching zero
over two, or just zero.

00:03:39.630 --> 00:03:42.053
That's approaching zero.

00:03:42.053 --> 00:03:46.319
One times zero, well this is
just going to be equal to zero.

00:03:46.319 --> 00:03:47.193
And we're done.

00:03:47.193 --> 00:03:49.650
Using that fact, and a little
bit of trig identities,

00:03:49.650 --> 00:03:51.308
and a little bit of
algebraic manipulation,

00:03:51.308 --> 00:03:53.514
we were able to show
that our original limit,

00:03:53.514 --> 00:03:54.772
the limit, as x approaches zero,

00:03:54.772 --> 00:03:58.958
of one minus cosine of x
over x is equal to zero.

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And I encourage you to graph it.

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You will see that that makes sense

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from a graphical point of view, as well.