0:00:00.534,0:00:03.773
- The differentiable[br]functions x and y are related

0:00:03.773,0:00:05.965
by the following equation.

0:00:05.965,0:00:08.715
The sine of x plus cosine of y

0:00:09.578,0:00:12.479
is equal to square root of two.

0:00:12.479,0:00:15.128
They also tell us that the derivative of x

0:00:15.128,0:00:17.897
with respect to t is equal to five.

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They also ask us find the derivative of y

0:00:21.530,0:00:25.145
with respect to t when y[br]is equal to pi over four

0:00:25.145,0:00:29.789
and zero is less than x[br]is less than pi over two.

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So given that they are[br]telling us the derivative

0:00:32.365,0:00:34.685
of x with respect to t and we wanna find

0:00:34.685,0:00:37.134
the derivative of y with respect to t,

0:00:37.134,0:00:41.301
it's a safe assumption that[br]both x and y are functions of t.

0:00:42.403,0:00:45.751
So you could even rewrite[br]this equation right over here.

0:00:45.751,0:00:48.584
You could rewrite it as sine of x,

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which is a function of t,

0:00:53.040,0:00:53.957
plus cosine

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of y, which is a function of t,

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is equal to square root of two.

0:01:02.920,0:01:04.542
Now, it might confuse you a little bit,

0:01:04.542,0:01:06.376
you're not used to seeing x as a function

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of a third variable or y as a function

0:01:08.524,0:01:10.068
of something other than x.

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But remember, x and y are just variables.

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This could be f of t,[br]and this could be g of t

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instead of x of t and y of t,

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and that might feel a[br]little more natural to you.

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But needless to say,[br]if we wanna find dy dt,

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what we want to do is take the derivative

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with respect to t of both[br]sides of this equation.

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So let's do that.

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So we're gonna do it[br]on the left-hand side,

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

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derivative of that with respect to t.

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

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And then we're gonna take the derivative

0:01:42.338,0:01:46.547
of the right-hand side, this[br]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.

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Let me do this in a new color.

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

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how could I write that?

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So I'm taking the derivative[br]with respect to t,

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I have sine of something, which[br]is itself a function of t.

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So I would just apply the chain rule here.

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I'm first going to take the[br]derivative with respect to x of

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sine of

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x, I could write sine of x of t,

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but I'll just revert back[br]to the sine of x here

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for simplicity.

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And then I will then multiply[br]that times the derivative

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of the inside, you could[br]say, with respect to t

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times the derivative[br]of x with respect to t.

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This might be a little counterintuitive

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to how you've applied[br]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,[br]I'm taking the derivative

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of the outside of the sine of something

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

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and then I'm taking the[br]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[br]derivative with respect to y

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of, I guess you could say the outside,

0:03:04.327,0:03:05.577
of cosine of y,

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and then I would multiply that

0:03:09.206,0:03:12.873
times the derivative[br]of y with respect to t.

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And then all of that is[br]going to be equal to what?

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Well, the derivative with[br]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,

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so its derivative, its[br]rate of change is zero.

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All right, so now we[br]just have to figure out

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all of these things.

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So first of all, the[br]derivative with respect to x

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of sine of x is cosine of[br]x times the derivative of x

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with respect to t, I'll[br]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,[br]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[br]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[br]cosine of y with respect to y?

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Well, that is negative sine of y.

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And so, actually let me[br]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?

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They've told us that the[br]derivative of x with respect to t

0:04:21.785,0:04:25.398
is equal to five, they tell[br]us that right over here.

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So this is equal to five.

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We wanna find the derivative[br]of y with respect to t.

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They tell us what y is, y is pi over four.

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This, y is pi over four, so[br]we know this is pi over four.

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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[br]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?

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Well, to figure that out,

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we can go back to this original[br]equation right over here.

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So when y is pi over four, you get,

0:05:03.754,0:05:04.847
let me write down.

0:05:04.847,0:05:05.680
Sine of x

0:05:07.391,0:05:08.808
plus cosine of pi

0:05:10.716,0:05:13.984
over four is equal to square root of two.

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Cosine of pi over four,

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we revert to our unit or we[br]think about our unit circle.

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We're in the first quadrant.

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If we think in degrees,[br]it's a 45 degree angle,

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that's gonna be square[br]root of two over two.

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And so we can subtract[br]square root of two over two

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from both sides, which is going to give us

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sine of x is equal to, well,[br]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[br]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[br]unit circle is going to be

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in that first quadrant, x[br]is an angle in this case

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right over here.

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

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So this tells us that x[br]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[br]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

0:06:17.521,0:06:19.354
cosine of pi over four

0:06:22.214,0:06:23.047
minus

0:06:24.268,0:06:26.968
dy dt, the derivative[br]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[br]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.

0:06:45.108,0:06:46.927
Cosine of pi over four,

0:06:46.927,0:06:49.677
we already know is square[br]root of two over two.

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Sine of pi over four is also[br]square root of two over two.

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

0:06:57.583,0:07:00.878
of this equation by square[br]root of two over two?

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Well, what's that gonna give us?

0:07:02.239,0:07:04.549
Well, then, this square[br]root of two over two

0:07:04.549,0:07:05.884
divided by square root of two over,

0:07:05.884,0:07:08.179
square root of two over two[br]divided square root of two

0:07:08.179,0:07:10.035
over two is gonna be one.

0:07:10.035,0:07:11.283
Square root of two over two divided

0:07:11.283,0:07:13.232
square root of two over[br]two is gonna be one.

0:07:13.232,0:07:15.495
And then zero divided by[br]square root of two over two

0:07:15.495,0:07:17.751
is just still going to be zero.

0:07:17.751,0:07:19.842
And so this whole thing simplifies to

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five times one, which is just five,

0:07:23.070,0:07:26.627
minus the derivative[br]of y with respect to t

0:07:26.627,0:07:28.044
is equal to zero,

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and so there you have it.

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

0:07:33.648,0:07:37.815
and we get the derivative of y[br]with respect to t is equal to

0:07:38.684,0:07:42.294
five, when all of these[br]other things are true.

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

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and the derivative and y, I should say,

0:07:47.999,0:07:50.166
is equal to pi over four.