[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.53,0:00:03.77,Default,,0000,0000,0000,,- The differentiable\Nfunctions x and y are related
Dialogue: 0,0:00:03.77,0:00:05.96,Default,,0000,0000,0000,,by the following equation.
Dialogue: 0,0:00:05.96,0:00:08.72,Default,,0000,0000,0000,,The sine of x plus cosine of y
Dialogue: 0,0:00:09.58,0:00:12.48,Default,,0000,0000,0000,,is equal to square root of two.
Dialogue: 0,0:00:12.48,0:00:15.13,Default,,0000,0000,0000,,They also tell us that the derivative of x
Dialogue: 0,0:00:15.13,0:00:17.90,Default,,0000,0000,0000,,with respect to t is equal to five.
Dialogue: 0,0:00:17.90,0:00:21.53,Default,,0000,0000,0000,,They also ask us find the derivative of y
Dialogue: 0,0:00:21.53,0:00:25.14,Default,,0000,0000,0000,,with respect to t when y\Nis equal to pi over four
Dialogue: 0,0:00:25.14,0:00:29.79,Default,,0000,0000,0000,,and zero is less than x\Nis less than pi over two.
Dialogue: 0,0:00:29.79,0:00:32.36,Default,,0000,0000,0000,,So given that they are\Ntelling us the derivative
Dialogue: 0,0:00:32.36,0:00:34.68,Default,,0000,0000,0000,,of x with respect to t and we wanna find
Dialogue: 0,0:00:34.68,0:00:37.13,Default,,0000,0000,0000,,the derivative of y with respect to t,
Dialogue: 0,0:00:37.13,0:00:41.30,Default,,0000,0000,0000,,it's a safe assumption that\Nboth x and y are functions of t.
Dialogue: 0,0:00:42.40,0:00:45.75,Default,,0000,0000,0000,,So you could even rewrite\Nthis equation right over here.
Dialogue: 0,0:00:45.75,0:00:48.58,Default,,0000,0000,0000,,You could rewrite it as sine of x,
Dialogue: 0,0:00:50.68,0:00:53.04,Default,,0000,0000,0000,,which is a function of t,
Dialogue: 0,0:00:53.04,0:00:53.96,Default,,0000,0000,0000,,plus cosine
Dialogue: 0,0:00:55.82,0:00:58.40,Default,,0000,0000,0000,,of y, which is a function of t,
Dialogue: 0,0:00:59.50,0:01:02.08,Default,,0000,0000,0000,,is equal to square root of two.
Dialogue: 0,0:01:02.92,0:01:04.54,Default,,0000,0000,0000,,Now, it might confuse you a little bit,
Dialogue: 0,0:01:04.54,0:01:06.38,Default,,0000,0000,0000,,you're not used to seeing x as a function
Dialogue: 0,0:01:06.38,0:01:08.52,Default,,0000,0000,0000,,of a third variable or y as a function
Dialogue: 0,0:01:08.52,0:01:10.07,Default,,0000,0000,0000,,of something other than x.
Dialogue: 0,0:01:10.07,0:01:11.74,Default,,0000,0000,0000,,But remember, x and y are just variables.
Dialogue: 0,0:01:11.74,0:01:15.42,Default,,0000,0000,0000,,This could be f of t,\Nand this could be g of t
Dialogue: 0,0:01:15.42,0:01:17.60,Default,,0000,0000,0000,,instead of x of t and y of t,
Dialogue: 0,0:01:17.60,0:01:19.50,Default,,0000,0000,0000,,and that might feel a\Nlittle more natural to you.
Dialogue: 0,0:01:19.50,0:01:23.17,Default,,0000,0000,0000,,But needless to say,\Nif we wanna find dy dt,
Dialogue: 0,0:01:24.19,0:01:26.26,Default,,0000,0000,0000,,what we want to do is take the derivative
Dialogue: 0,0:01:26.26,0:01:29.95,Default,,0000,0000,0000,,with respect to t of both\Nsides of this equation.
Dialogue: 0,0:01:29.95,0:01:31.36,Default,,0000,0000,0000,,So let's do that.
Dialogue: 0,0:01:31.36,0:01:33.30,Default,,0000,0000,0000,,So we're gonna do it\Non the left-hand side,
Dialogue: 0,0:01:33.30,0:01:36.54,Default,,0000,0000,0000,,so it's gonna be we take\Nthat with respect to t,
Dialogue: 0,0:01:36.54,0:01:38.04,Default,,0000,0000,0000,,derivative of that with respect to t.
Dialogue: 0,0:01:38.04,0:01:41.00,Default,,0000,0000,0000,,We're gonna take the derivative\Nof that with respect to t.
Dialogue: 0,0:01:41.00,0:01:42.34,Default,,0000,0000,0000,,And then we're gonna take the derivative
Dialogue: 0,0:01:42.34,0:01:46.55,Default,,0000,0000,0000,,of the right-hand side, this\Nconstant with respect to t.
Dialogue: 0,0:01:46.55,0:01:49.76,Default,,0000,0000,0000,,So let's think about each of these things.
Dialogue: 0,0:01:49.76,0:01:51.44,Default,,0000,0000,0000,,So what is this.
Dialogue: 0,0:01:51.44,0:01:53.11,Default,,0000,0000,0000,,Let me do this in a new color.
Dialogue: 0,0:01:53.11,0:01:56.62,Default,,0000,0000,0000,,The stuff that I'm doing in\Nthis aqua color right over here,
Dialogue: 0,0:01:56.62,0:01:58.24,Default,,0000,0000,0000,,how could I write that?
Dialogue: 0,0:01:58.24,0:02:00.40,Default,,0000,0000,0000,,So I'm taking the derivative\Nwith respect to t,
Dialogue: 0,0:02:00.40,0:02:04.92,Default,,0000,0000,0000,,I have sine of something, which\Nis itself a function of t.
Dialogue: 0,0:02:04.92,0:02:07.77,Default,,0000,0000,0000,,So I would just apply the chain rule here.
Dialogue: 0,0:02:07.77,0:02:11.94,Default,,0000,0000,0000,,I'm first going to take the\Nderivative with respect to x of
Dialogue: 0,0:02:13.82,0:02:14.65,Default,,0000,0000,0000,,sine of
Dialogue: 0,0:02:16.51,0:02:18.71,Default,,0000,0000,0000,,x, I could write sine of x of t,
Dialogue: 0,0:02:18.71,0:02:20.88,Default,,0000,0000,0000,,but I'll just revert back\Nto the sine of x here
Dialogue: 0,0:02:20.88,0:02:22.36,Default,,0000,0000,0000,,for simplicity.
Dialogue: 0,0:02:22.36,0:02:25.24,Default,,0000,0000,0000,,And then I will then multiply\Nthat times the derivative
Dialogue: 0,0:02:25.24,0:02:28.77,Default,,0000,0000,0000,,of the inside, you could\Nsay, with respect to t
Dialogue: 0,0:02:28.77,0:02:32.78,Default,,0000,0000,0000,,times the derivative\Nof x with respect to t.
Dialogue: 0,0:02:32.78,0:02:34.51,Default,,0000,0000,0000,,This might be a little counterintuitive
Dialogue: 0,0:02:34.51,0:02:36.68,Default,,0000,0000,0000,,to how you've applied\Nthe chain rule before
Dialogue: 0,0:02:36.68,0:02:38.74,Default,,0000,0000,0000,,when we only dealt with xs and ys,
Dialogue: 0,0:02:38.74,0:02:41.27,Default,,0000,0000,0000,,but all that's happening,\NI'm taking the derivative
Dialogue: 0,0:02:41.27,0:02:43.56,Default,,0000,0000,0000,,of the outside of the sine of something
Dialogue: 0,0:02:43.56,0:02:46.55,Default,,0000,0000,0000,,with respect to the something,\Nin this case, it is x,
Dialogue: 0,0:02:46.55,0:02:48.50,Default,,0000,0000,0000,,and then I'm taking the\Nderivative of the something,
Dialogue: 0,0:02:48.50,0:02:51.42,Default,,0000,0000,0000,,in this case, x with respect to t.
Dialogue: 0,0:02:51.42,0:02:53.93,Default,,0000,0000,0000,,Well, we can do the same thing here,
Dialogue: 0,0:02:53.93,0:02:56.01,Default,,0000,0000,0000,,or this second term here.
Dialogue: 0,0:02:56.99,0:03:01.22,Default,,0000,0000,0000,,So I wanna take the\Nderivative with respect to y
Dialogue: 0,0:03:01.22,0:03:04.33,Default,,0000,0000,0000,,of, I guess you could say the outside,
Dialogue: 0,0:03:04.33,0:03:05.58,Default,,0000,0000,0000,,of cosine of y,
Dialogue: 0,0:03:07.69,0:03:09.21,Default,,0000,0000,0000,,and then I would multiply that
Dialogue: 0,0:03:09.21,0:03:12.87,Default,,0000,0000,0000,,times the derivative\Nof y with respect to t.
Dialogue: 0,0:03:14.26,0:03:17.45,Default,,0000,0000,0000,,And then all of that is\Ngoing to be equal to what?
Dialogue: 0,0:03:17.45,0:03:20.74,Default,,0000,0000,0000,,Well, the derivative with\Nrespect to t of a constant,
Dialogue: 0,0:03:20.74,0:03:22.16,Default,,0000,0000,0000,,square root of two is a constant,
Dialogue: 0,0:03:22.16,0:03:23.91,Default,,0000,0000,0000,,it's not gonna change as t changes,
Dialogue: 0,0:03:23.91,0:03:27.38,Default,,0000,0000,0000,,so its derivative, its\Nrate of change is zero.
Dialogue: 0,0:03:27.38,0:03:29.63,Default,,0000,0000,0000,,All right, so now we\Njust have to figure out
Dialogue: 0,0:03:29.63,0:03:31.36,Default,,0000,0000,0000,,all of these things.
Dialogue: 0,0:03:31.36,0:03:33.68,Default,,0000,0000,0000,,So first of all, the\Nderivative with respect to x
Dialogue: 0,0:03:33.68,0:03:38.28,Default,,0000,0000,0000,,of sine of x is cosine of\Nx times the derivative of x
Dialogue: 0,0:03:38.28,0:03:40.27,Default,,0000,0000,0000,,with respect to t, I'll\Njust write that out here.
Dialogue: 0,0:03:40.27,0:03:42.21,Default,,0000,0000,0000,,The derivative of x with respect to t.
Dialogue: 0,0:03:42.21,0:03:44.96,Default,,0000,0000,0000,,And then we're going to have,\Nit's gonna be a plus here,
Dialogue: 0,0:03:44.96,0:03:47.16,Default,,0000,0000,0000,,the derivative of y with respect to t.
Dialogue: 0,0:03:47.16,0:03:51.01,Default,,0000,0000,0000,,So plus the derivative\Nof y with respect to t.
Dialogue: 0,0:03:51.01,0:03:52.44,Default,,0000,0000,0000,,I'm just flopping the order here,
Dialogue: 0,0:03:52.44,0:03:54.47,Default,,0000,0000,0000,,so that this goes out front.
Dialogue: 0,0:03:54.47,0:03:58.37,Default,,0000,0000,0000,,Now, what's the derivative of\Ncosine of y with respect to y?
Dialogue: 0,0:03:58.37,0:04:01.34,Default,,0000,0000,0000,,Well, that is negative sine of y.
Dialogue: 0,0:04:01.34,0:04:05.26,Default,,0000,0000,0000,,And so, actually let me\Njust put a sine of y here,
Dialogue: 0,0:04:05.26,0:04:07.12,Default,,0000,0000,0000,,then I'm gonna have a negative.
Dialogue: 0,0:04:07.12,0:04:10.12,Default,,0000,0000,0000,,Erase this and put a negative there.
Dialogue: 0,0:04:11.60,0:04:15.10,Default,,0000,0000,0000,,And that is all going to be equal to zero.
Dialogue: 0,0:04:16.06,0:04:18.74,Default,,0000,0000,0000,,And so what can we figure out now?
Dialogue: 0,0:04:18.74,0:04:21.78,Default,,0000,0000,0000,,They've told us that the\Nderivative of x with respect to t
Dialogue: 0,0:04:21.78,0:04:25.40,Default,,0000,0000,0000,,is equal to five, they tell\Nus that right over here.
Dialogue: 0,0:04:25.40,0:04:27.48,Default,,0000,0000,0000,,So this is equal to five.
Dialogue: 0,0:04:29.09,0:04:32.68,Default,,0000,0000,0000,,We wanna find the derivative\Nof y with respect to t.
Dialogue: 0,0:04:32.68,0:04:36.14,Default,,0000,0000,0000,,They tell us what y is, y is pi over four.
Dialogue: 0,0:04:36.14,0:04:40.31,Default,,0000,0000,0000,,This, y is pi over four, so\Nwe know this is pi over four.
Dialogue: 0,0:04:41.61,0:04:43.79,Default,,0000,0000,0000,,And let's see, we have to figure out what,
Dialogue: 0,0:04:43.79,0:04:45.72,Default,,0000,0000,0000,,we still have two unknowns here.
Dialogue: 0,0:04:45.72,0:04:47.45,Default,,0000,0000,0000,,We don't know what x is and we don't know
Dialogue: 0,0:04:47.45,0:04:49.58,Default,,0000,0000,0000,,what the derivative of\Ny with respect to t is.
Dialogue: 0,0:04:49.58,0:04:51.10,Default,,0000,0000,0000,,This is what we need to figure out.
Dialogue: 0,0:04:51.10,0:04:52.47,Default,,0000,0000,0000,,So what would x be?
Dialogue: 0,0:04:52.47,0:04:55.42,Default,,0000,0000,0000,,What would x be when y is pi over four?
Dialogue: 0,0:04:55.42,0:04:56.44,Default,,0000,0000,0000,,Well, to figure that out,
Dialogue: 0,0:04:56.44,0:05:00.22,Default,,0000,0000,0000,,we can go back to this original\Nequation right over here.
Dialogue: 0,0:05:00.22,0:05:03.75,Default,,0000,0000,0000,,So when y is pi over four, you get,
Dialogue: 0,0:05:03.75,0:05:04.85,Default,,0000,0000,0000,,let me write down.
Dialogue: 0,0:05:04.85,0:05:05.68,Default,,0000,0000,0000,,Sine of x
Dialogue: 0,0:05:07.39,0:05:08.81,Default,,0000,0000,0000,,plus cosine of pi
Dialogue: 0,0:05:10.72,0:05:13.98,Default,,0000,0000,0000,,over four is equal to square root of two.
Dialogue: 0,0:05:13.98,0:05:15.90,Default,,0000,0000,0000,,Cosine of pi over four,
Dialogue: 0,0:05:17.59,0:05:20.94,Default,,0000,0000,0000,,we revert to our unit or we\Nthink about our unit circle.
Dialogue: 0,0:05:20.94,0:05:22.56,Default,,0000,0000,0000,,We're in the first quadrant.
Dialogue: 0,0:05:22.56,0:05:24.16,Default,,0000,0000,0000,,If we think in degrees,\Nit's a 45 degree angle,
Dialogue: 0,0:05:24.16,0:05:28.07,Default,,0000,0000,0000,,that's gonna be square\Nroot of two over two.
Dialogue: 0,0:05:28.07,0:05:30.58,Default,,0000,0000,0000,,And so we can subtract\Nsquare root of two over two
Dialogue: 0,0:05:30.58,0:05:32.85,Default,,0000,0000,0000,,from both sides, which is going to give us
Dialogue: 0,0:05:32.85,0:05:37.71,Default,,0000,0000,0000,,sine of x is equal to, well,\Nif you take square root of two
Dialogue: 0,0:05:37.71,0:05:39.47,Default,,0000,0000,0000,,over two from square root of two,
Dialogue: 0,0:05:39.47,0:05:40.76,Default,,0000,0000,0000,,you're taking half of it away,
Dialogue: 0,0:05:40.76,0:05:42.22,Default,,0000,0000,0000,,so you're gonna have half of it left.
Dialogue: 0,0:05:42.22,0:05:44.71,Default,,0000,0000,0000,,So square root of two over two.
Dialogue: 0,0:05:44.71,0:05:48.72,Default,,0000,0000,0000,,And so, what x value, when\NI take the sine of it,
Dialogue: 0,0:05:48.72,0:05:50.77,Default,,0000,0000,0000,,and remember, where the angle,
Dialogue: 0,0:05:50.77,0:05:52.36,Default,,0000,0000,0000,,if we're thinking when the\Nunit circle is going to be
Dialogue: 0,0:05:52.36,0:05:54.78,Default,,0000,0000,0000,,in that first quadrant, x\Nis an angle in this case
Dialogue: 0,0:05:54.78,0:05:56.08,Default,,0000,0000,0000,,right over here.
Dialogue: 0,0:05:56.08,0:05:59.38,Default,,0000,0000,0000,,Well, that's going to be\Nonce again pi over four.
Dialogue: 0,0:05:59.38,0:06:03.16,Default,,0000,0000,0000,,So this tells us that x\Nis equal to pi over four
Dialogue: 0,0:06:03.16,0:06:05.83,Default,,0000,0000,0000,,when y is equal to pi over four.
Dialogue: 0,0:06:05.83,0:06:09.48,Default,,0000,0000,0000,,And so we know that this\Nis pi over four as well.
Dialogue: 0,0:06:09.48,0:06:11.44,Default,,0000,0000,0000,,So let me just rewrite this,
Dialogue: 0,0:06:11.44,0:06:13.46,Default,,0000,0000,0000,,because it's getting a little bit messy.
Dialogue: 0,0:06:13.46,0:06:15.63,Default,,0000,0000,0000,,So we know that five times
Dialogue: 0,0:06:17.52,0:06:19.35,Default,,0000,0000,0000,,cosine of pi over four
Dialogue: 0,0:06:22.21,0:06:23.05,Default,,0000,0000,0000,,minus
Dialogue: 0,0:06:24.27,0:06:26.97,Default,,0000,0000,0000,,dy dt, the derivative\Nof y with respect to t,
Dialogue: 0,0:06:26.97,0:06:28.77,Default,,0000,0000,0000,,which is what we want to figure out,
Dialogue: 0,0:06:28.77,0:06:31.02,Default,,0000,0000,0000,,times sine of pi over four,
Dialogue: 0,0:06:32.56,0:06:33.98,Default,,0000,0000,0000,,is equal to zero,
Dialogue: 0,0:06:35.40,0:06:38.52,Default,,0000,0000,0000,,is equal to zero, and we\Nput some parentheses here,
Dialogue: 0,0:06:38.52,0:06:40.71,Default,,0000,0000,0000,,just to clarify things a little bit.
Dialogue: 0,0:06:40.71,0:06:43.45,Default,,0000,0000,0000,,All right, so let's see.
Dialogue: 0,0:06:43.45,0:06:45.11,Default,,0000,0000,0000,,Now, it's just a little bit of algebra.
Dialogue: 0,0:06:45.11,0:06:46.93,Default,,0000,0000,0000,,Cosine of pi over four,
Dialogue: 0,0:06:46.93,0:06:49.68,Default,,0000,0000,0000,,we already know is square\Nroot of two over two.
Dialogue: 0,0:06:49.68,0:06:53.84,Default,,0000,0000,0000,,Sine of pi over four is also\Nsquare root of two over two.
Dialogue: 0,0:06:54.75,0:06:57.58,Default,,0000,0000,0000,,Now let's see, what if\Nwe divide both sides
Dialogue: 0,0:06:57.58,0:07:00.88,Default,,0000,0000,0000,,of this equation by square\Nroot of two over two?
Dialogue: 0,0:07:00.88,0:07:02.24,Default,,0000,0000,0000,,Well, what's that gonna give us?
Dialogue: 0,0:07:02.24,0:07:04.55,Default,,0000,0000,0000,,Well, then, this square\Nroot of two over two
Dialogue: 0,0:07:04.55,0:07:05.88,Default,,0000,0000,0000,,divided by square root of two over,
Dialogue: 0,0:07:05.88,0:07:08.18,Default,,0000,0000,0000,,square root of two over two\Ndivided square root of two
Dialogue: 0,0:07:08.18,0:07:10.04,Default,,0000,0000,0000,,over two is gonna be one.
Dialogue: 0,0:07:10.04,0:07:11.28,Default,,0000,0000,0000,,Square root of two over two divided
Dialogue: 0,0:07:11.28,0:07:13.23,Default,,0000,0000,0000,,square root of two over\Ntwo is gonna be one.
Dialogue: 0,0:07:13.23,0:07:15.50,Default,,0000,0000,0000,,And then zero divided by\Nsquare root of two over two
Dialogue: 0,0:07:15.50,0:07:17.75,Default,,0000,0000,0000,,is just still going to be zero.
Dialogue: 0,0:07:17.75,0:07:19.84,Default,,0000,0000,0000,,And so this whole thing simplifies to
Dialogue: 0,0:07:19.84,0:07:23.07,Default,,0000,0000,0000,,five times one, which is just five,
Dialogue: 0,0:07:23.07,0:07:26.63,Default,,0000,0000,0000,,minus the derivative\Nof y with respect to t
Dialogue: 0,0:07:26.63,0:07:28.04,Default,,0000,0000,0000,,is equal to zero,
Dialogue: 0,0:07:29.56,0:07:30.73,Default,,0000,0000,0000,,and so there you have it.
Dialogue: 0,0:07:30.73,0:07:33.65,Default,,0000,0000,0000,,You add the derivative of y\Nwith respect to t to both sides,
Dialogue: 0,0:07:33.65,0:07:37.82,Default,,0000,0000,0000,,and we get the derivative of y\Nwith respect to t is equal to
Dialogue: 0,0:07:38.68,0:07:42.29,Default,,0000,0000,0000,,five, when all of these\Nother things are true.
Dialogue: 0,0:07:42.29,0:07:44.66,Default,,0000,0000,0000,,When the derivative of x\Nwith respect to t is five,
Dialogue: 0,0:07:44.66,0:07:47.100,Default,,0000,0000,0000,,and the derivative and y, I should say,
Dialogue: 0,0:07:47.100,0:07:50.17,Default,,0000,0000,0000,,is equal to pi over four.