[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.00,0:00:00.43,Default,,0000,0000,0000,,
Dialogue: 0,0:00:00.43,0:00:04.05,Default,,0000,0000,0000,,I've been asked implicitly\Ndifferentiate the equation
Dialogue: 0,0:00:04.05,0:00:10.39,Default,,0000,0000,0000,,tangent of x over y is\Nequal to x plus y.
Dialogue: 0,0:00:10.39,0:00:14.15,Default,,0000,0000,0000,,And I've done several implicit\Ndifferentiation videos, but
Dialogue: 0,0:00:14.15,0:00:17.44,Default,,0000,0000,0000,,this tends to be one of the\Nbiggest sources of pain for
Dialogue: 0,0:00:17.44,0:00:18.72,Default,,0000,0000,0000,,first year calculus students.
Dialogue: 0,0:00:18.72,0:00:21.04,Default,,0000,0000,0000,,So I thought I would give\Nat least another example.
Dialogue: 0,0:00:21.04,0:00:22.86,Default,,0000,0000,0000,,It never hurts to see\Nas many as possible.
Dialogue: 0,0:00:22.86,0:00:24.29,Default,,0000,0000,0000,,So let's do this one.
Dialogue: 0,0:00:24.29,0:00:26.68,Default,,0000,0000,0000,,So to implicitly differentiate\Nthis, we just apply the
Dialogue: 0,0:00:26.68,0:00:29.36,Default,,0000,0000,0000,,derivative with respect to x\Noperator to both sides
Dialogue: 0,0:00:29.36,0:00:29.97,Default,,0000,0000,0000,,of the equation.
Dialogue: 0,0:00:29.97,0:00:33.29,Default,,0000,0000,0000,,The derivative with this\Nrespect to x -- the derivative
Dialogue: 0,0:00:33.29,0:00:35.42,Default,,0000,0000,0000,,of the left side with respect\Nto x is the same as the
Dialogue: 0,0:00:35.42,0:00:40.58,Default,,0000,0000,0000,,derivative of the right\Nside with respect to x.
Dialogue: 0,0:00:40.58,0:00:42.79,Default,,0000,0000,0000,,The right side's going to be\Nvery straightforward, but
Dialogue: 0,0:00:42.79,0:00:44.77,Default,,0000,0000,0000,,the left side is a\Nlittle bit tricky.
Dialogue: 0,0:00:44.77,0:00:47.38,Default,,0000,0000,0000,,So let's do that on\Nthe side over here.
Dialogue: 0,0:00:47.38,0:00:52.02,Default,,0000,0000,0000,,Let me write the left hand side\Na little bit differently.
Dialogue: 0,0:00:52.02,0:00:52.99,Default,,0000,0000,0000,,I'm going to do it in\Na different color.
Dialogue: 0,0:00:52.99,0:01:00.41,Default,,0000,0000,0000,,Let me say that a is equal\Nto the tangent of b.
Dialogue: 0,0:01:00.41,0:01:09.38,Default,,0000,0000,0000,,And let me say that b\Nis equal to x over y.
Dialogue: 0,0:01:09.38,0:01:11.62,Default,,0000,0000,0000,,Then a is clearly\Nthe same thing.
Dialogue: 0,0:01:11.62,0:01:14.86,Default,,0000,0000,0000,,I mean if I just substituted\Nb back in here, a, this
Dialogue: 0,0:01:14.86,0:01:18.09,Default,,0000,0000,0000,,whole thing I could\Nre-write as just a.
Dialogue: 0,0:01:18.09,0:01:20.93,Default,,0000,0000,0000,,So if we're taking the\Nderivative of a with respect
Dialogue: 0,0:01:20.93,0:01:23.74,Default,,0000,0000,0000,,to x, that's what we\Nwant to do right here.
Dialogue: 0,0:01:23.74,0:01:26.57,Default,,0000,0000,0000,,Let me just take the derivative\Nof both sides of this.
Dialogue: 0,0:01:26.57,0:01:36.50,Default,,0000,0000,0000,,This would be the derivative of\Na with respect to x is equal to
Dialogue: 0,0:01:36.50,0:01:38.61,Default,,0000,0000,0000,,the derivative of x\Nwith respect to x.
Dialogue: 0,0:01:38.61,0:01:41.21,Default,,0000,0000,0000,,Well that's pretty\Nstraightforward, that's just 1.
Dialogue: 0,0:01:41.21,0:01:44.39,Default,,0000,0000,0000,,Plus the derivative of\Ny with respect to x.
Dialogue: 0,0:01:44.39,0:01:45.43,Default,,0000,0000,0000,,So let me write it like this.
Dialogue: 0,0:01:45.43,0:01:48.82,Default,,0000,0000,0000,,I'll write the derivative\Noperator, the derivative
Dialogue: 0,0:01:48.82,0:01:53.77,Default,,0000,0000,0000,,oh y with respect to x.
Dialogue: 0,0:01:53.77,0:01:54.35,Default,,0000,0000,0000,,That's all we did.
Dialogue: 0,0:01:54.35,0:01:56.52,Default,,0000,0000,0000,,We just applied the derivative\Noperator to y, and we don't
Dialogue: 0,0:01:56.52,0:01:58.65,Default,,0000,0000,0000,,know what this thing is,\Nwe're going to solve for it.
Dialogue: 0,0:01:58.65,0:02:01.18,Default,,0000,0000,0000,,But obviously, I can't just\Nleave this here, the derivative
Dialogue: 0,0:02:01.18,0:02:02.36,Default,,0000,0000,0000,,of a with respect to x.
Dialogue: 0,0:02:02.36,0:02:04.61,Default,,0000,0000,0000,,We just solved for a, and\Na is just this thing
Dialogue: 0,0:02:04.61,0:02:05.93,Default,,0000,0000,0000,,right here, right?
Dialogue: 0,0:02:05.93,0:02:09.45,Default,,0000,0000,0000,,a is tangent of b, and\Nb is just y over x.
Dialogue: 0,0:02:09.45,0:02:11.73,Default,,0000,0000,0000,,The reason why I wrote it this\Nway is because I wanted to show
Dialogue: 0,0:02:11.73,0:02:14.87,Default,,0000,0000,0000,,you that when you take the\Nderivative of this, it just
Dialogue: 0,0:02:14.87,0:02:16.50,Default,,0000,0000,0000,,comes out of the chain rule.
Dialogue: 0,0:02:16.50,0:02:18.84,Default,,0000,0000,0000,,It's not some type of new\Nvoodoo magic that you
Dialogue: 0,0:02:18.84,0:02:20.09,Default,,0000,0000,0000,,haven't learned yet.
Dialogue: 0,0:02:20.09,0:02:22.20,Default,,0000,0000,0000,,So the derivative -- let\Nme just write down the
Dialogue: 0,0:02:22.20,0:02:23.99,Default,,0000,0000,0000,,chain rule right here.
Dialogue: 0,0:02:23.99,0:02:30.93,Default,,0000,0000,0000,,The derivative of a with\Nrespect to x is equal to the
Dialogue: 0,0:02:30.93,0:02:35.28,Default,,0000,0000,0000,,derivative of a with respect\Nto b times the derivative
Dialogue: 0,0:02:35.28,0:02:37.58,Default,,0000,0000,0000,,of b with respect to x.
Dialogue: 0,0:02:37.58,0:02:39.72,Default,,0000,0000,0000,,That's just the chain rule and\Nit's very easy to remember,
Dialogue: 0,0:02:39.72,0:02:43.04,Default,,0000,0000,0000,,because the db's cancel out and\Nyou're just left with the
Dialogue: 0,0:02:43.04,0:02:45.80,Default,,0000,0000,0000,,derivative of a with respect to\Nx, if you just treated these
Dialogue: 0,0:02:45.80,0:02:47.47,Default,,0000,0000,0000,,like regular fractions.
Dialogue: 0,0:02:47.47,0:02:50.28,Default,,0000,0000,0000,,So what's the derivative\Nof a with respect to b?
Dialogue: 0,0:02:50.28,0:02:55.02,Default,,0000,0000,0000,,
Dialogue: 0,0:02:55.02,0:03:01.57,Default,,0000,0000,0000,,Well, that's just 1 over\Ncosine squared of b.
Dialogue: 0,0:03:01.57,0:03:03.57,Default,,0000,0000,0000,,And if you don't have that\Nmemorized, it's actually not
Dialogue: 0,0:03:03.57,0:03:07.40,Default,,0000,0000,0000,,too hard to prove to yourself\Nif you just write this as sine
Dialogue: 0,0:03:07.40,0:03:10.67,Default,,0000,0000,0000,,of b over cosine of b, but this\Ntends to be one of the trig
Dialogue: 0,0:03:10.67,0:03:12.13,Default,,0000,0000,0000,,derivatives that most\Npeople memorize.
Dialogue: 0,0:03:12.13,0:03:14.23,Default,,0000,0000,0000,,I think I've already made a\Nvideo where I proved this.
Dialogue: 0,0:03:14.23,0:03:16.84,Default,,0000,0000,0000,,And some books still write this\Nas secant squared of b, but we
Dialogue: 0,0:03:16.84,0:03:19.07,Default,,0000,0000,0000,,know that secant squared is the\Nsame thing as 1 over
Dialogue: 0,0:03:19.07,0:03:20.34,Default,,0000,0000,0000,,cosine squared.
Dialogue: 0,0:03:20.34,0:03:25.32,Default,,0000,0000,0000,,I like to keep it in kind of\Nthe fundamental trig functions,
Dialogue: 0,0:03:25.32,0:03:27.36,Default,,0000,0000,0000,,or trig ratios as opposed to\Nthings like secant
Dialogue: 0,0:03:27.36,0:03:28.49,Default,,0000,0000,0000,,and cosecant.
Dialogue: 0,0:03:28.49,0:03:31.09,Default,,0000,0000,0000,,Then what's the derivative\Nof b with respect to x?
Dialogue: 0,0:03:31.09,0:03:37.03,Default,,0000,0000,0000,,
Dialogue: 0,0:03:37.03,0:03:38.26,Default,,0000,0000,0000,,So this is pretty interesting.
Dialogue: 0,0:03:38.26,0:03:39.71,Default,,0000,0000,0000,,Let me re-write b, actually.
Dialogue: 0,0:03:39.71,0:03:45.73,Default,,0000,0000,0000,,Let me write b is equal to\Nx times y to the minus 1.
Dialogue: 0,0:03:45.73,0:03:48.52,Default,,0000,0000,0000,,So the derivative of b with\Nrespect to x, we could do a
Dialogue: 0,0:03:48.52,0:03:50.47,Default,,0000,0000,0000,,little bit of chain\Nrule right here.
Dialogue: 0,0:03:50.47,0:03:53.68,Default,,0000,0000,0000,,We could say -- let me write\Nthis -- the derivative of b
Dialogue: 0,0:03:53.68,0:03:57.53,Default,,0000,0000,0000,,with respect to x is equal to\Nthe derivative of x times
Dialogue: 0,0:03:57.53,0:03:58.79,Default,,0000,0000,0000,,y to the negative 1.
Dialogue: 0,0:03:58.79,0:04:01.30,Default,,0000,0000,0000,,So the derivative of x is 1.
Dialogue: 0,0:04:01.30,0:04:07.36,Default,,0000,0000,0000,,times y to the negative 1 plus\Nthe derivative of y -- so
Dialogue: 0,0:04:07.36,0:04:08.03,Default,,0000,0000,0000,,let me just write this.
Dialogue: 0,0:04:08.03,0:04:12.32,Default,,0000,0000,0000,,Plus the derivative with\Nrespect to x of y to
Dialogue: 0,0:04:12.32,0:04:17.93,Default,,0000,0000,0000,,the minus 1 times the\Nfirst term, times x.
Dialogue: 0,0:04:17.93,0:04:20.47,Default,,0000,0000,0000,,So this thing right here, and\Nclearly I haven't completely
Dialogue: 0,0:04:20.47,0:04:21.19,Default,,0000,0000,0000,,simplified it yet.
Dialogue: 0,0:04:21.19,0:04:22.89,Default,,0000,0000,0000,,I still have to figure out\Nwhat this thing is here.
Dialogue: 0,0:04:22.89,0:04:25.01,Default,,0000,0000,0000,,But I just simply applied\Nthe product rule here.
Dialogue: 0,0:04:25.01,0:04:27.99,Default,,0000,0000,0000,,Your derivative of the first\Nterm, derivative of x is 1
Dialogue: 0,0:04:27.99,0:04:30.38,Default,,0000,0000,0000,,times the second term plus\Nthe derivative of the second
Dialogue: 0,0:04:30.38,0:04:31.31,Default,,0000,0000,0000,,term times the first term.
Dialogue: 0,0:04:31.31,0:04:32.70,Default,,0000,0000,0000,,That's all I did there.
Dialogue: 0,0:04:32.70,0:04:35.17,Default,,0000,0000,0000,,So the derivative of b\Nwith respect to x is just
Dialogue: 0,0:04:35.17,0:04:36.56,Default,,0000,0000,0000,,this thing right there.
Dialogue: 0,0:04:36.56,0:04:42.29,Default,,0000,0000,0000,,So it equals -- let me do it in\Nthe yellow -- so it's times --
Dialogue: 0,0:04:42.29,0:04:43.52,Default,,0000,0000,0000,,oh, I'll do it in the blue\Nsince I already wrote it.
Dialogue: 0,0:04:43.52,0:04:47.29,Default,,0000,0000,0000,,This is the blue, derivative of\Nb with respect to x is y to the
Dialogue: 0,0:04:47.29,0:04:52.58,Default,,0000,0000,0000,,minus 1, or 1 over y plus the\Nderivative with respect to
Dialogue: 0,0:04:52.58,0:04:59.59,Default,,0000,0000,0000,,x of 1 over y times x.
Dialogue: 0,0:04:59.59,0:05:01.18,Default,,0000,0000,0000,,So let me write that down here.
Dialogue: 0,0:05:01.18,0:05:04.33,Default,,0000,0000,0000,,So we just figured out, or\Nwe're almost done figuring
Dialogue: 0,0:05:04.33,0:05:07.40,Default,,0000,0000,0000,,out, what the derivative of a\Nwith respect to x is, and we
Dialogue: 0,0:05:07.40,0:05:08.45,Default,,0000,0000,0000,,could throw that in there.
Dialogue: 0,0:05:08.45,0:05:09.23,Default,,0000,0000,0000,,But we're not done.
Dialogue: 0,0:05:09.23,0:05:12.28,Default,,0000,0000,0000,,What's the derivative of 1\Nover y with respect to x?
Dialogue: 0,0:05:12.28,0:05:14.99,Default,,0000,0000,0000,,Well, do the chain rule again.
Dialogue: 0,0:05:14.99,0:05:17.52,Default,,0000,0000,0000,,
Dialogue: 0,0:05:17.52,0:05:18.83,Default,,0000,0000,0000,,And I want to be very\Nclear with this.
Dialogue: 0,0:05:18.83,0:05:21.57,Default,,0000,0000,0000,,I know this might seem a little\Nbit cumbersome what I'm doing
Dialogue: 0,0:05:21.57,0:05:24.02,Default,,0000,0000,0000,,here, but I think it might\Nmake a little bit of sense.
Dialogue: 0,0:05:24.02,0:05:28.39,Default,,0000,0000,0000,,Let me just set c is\Nequal to 1 over y.
Dialogue: 0,0:05:28.39,0:05:32.55,Default,,0000,0000,0000,,So the derivative of c with\Nrespect to x, just from the
Dialogue: 0,0:05:32.55,0:05:35.58,Default,,0000,0000,0000,,chain rule, is equal to the\Nderivative of c with respect
Dialogue: 0,0:05:35.58,0:05:40.09,Default,,0000,0000,0000,,to y times the derivative\Nof y with respect to x.
Dialogue: 0,0:05:40.09,0:05:43.14,Default,,0000,0000,0000,,What's the derivative of\Nc with respect to y?
Dialogue: 0,0:05:43.14,0:05:44.93,Default,,0000,0000,0000,,Well this is the same thing\Nas -- I could re-write
Dialogue: 0,0:05:44.93,0:05:46.35,Default,,0000,0000,0000,,this as y to the minus 1.
Dialogue: 0,0:05:46.35,0:05:51.16,Default,,0000,0000,0000,,So it's minus y to\Nthe minus 2 power.
Dialogue: 0,0:05:51.16,0:05:52.91,Default,,0000,0000,0000,,That's what this thing is.
Dialogue: 0,0:05:52.91,0:05:55.74,Default,,0000,0000,0000,,This thing is that right there.
Dialogue: 0,0:05:55.74,0:05:57.22,Default,,0000,0000,0000,,And I don't know what the\Nderivative of y with
Dialogue: 0,0:05:57.22,0:05:58.02,Default,,0000,0000,0000,,respect to x is.
Dialogue: 0,0:05:58.02,0:05:59.69,Default,,0000,0000,0000,,That's what we're\Ntrying to solve for.
Dialogue: 0,0:05:59.69,0:06:02.39,Default,,0000,0000,0000,,So it's that times\Nthe derivative of y
Dialogue: 0,0:06:02.39,0:06:03.54,Default,,0000,0000,0000,,with respect to x.
Dialogue: 0,0:06:03.54,0:06:05.34,Default,,0000,0000,0000,,That just comes out\Nof the chain rule.
Dialogue: 0,0:06:05.34,0:06:11.40,Default,,0000,0000,0000,,So this thing right here, this\Nis the derivative of this thing
Dialogue: 0,0:06:11.40,0:06:13.83,Default,,0000,0000,0000,,with respect to x, which is the\Nsame thing as derivative
Dialogue: 0,0:06:13.83,0:06:15.77,Default,,0000,0000,0000,,of c with respect to x.
Dialogue: 0,0:06:15.77,0:06:19.21,Default,,0000,0000,0000,,So I can write this little\Npiece right here, I can
Dialogue: 0,0:06:19.21,0:06:25.24,Default,,0000,0000,0000,,re-write this little piece as\Nminus y to the minus 2 dy
Dialogue: 0,0:06:25.24,0:06:28.91,Default,,0000,0000,0000,,dx, and then, of course,\Nthat there is times x.
Dialogue: 0,0:06:28.91,0:06:33.91,Default,,0000,0000,0000,,And then we had the plus 1 over\Ny, and all of that was times
Dialogue: 0,0:06:33.91,0:06:38.05,Default,,0000,0000,0000,,the 1 over cosine squared of b.
Dialogue: 0,0:06:38.05,0:06:40.66,Default,,0000,0000,0000,,So now we've simplified\Nthis a good bit.
Dialogue: 0,0:06:40.66,0:06:42.84,Default,,0000,0000,0000,,I hope going into the chain\Nrule didn't confuse you,
Dialogue: 0,0:06:42.84,0:06:45.02,Default,,0000,0000,0000,,because I really want to hit\Nthe point home that all of
Dialogue: 0,0:06:45.02,0:06:48.32,Default,,0000,0000,0000,,these implicit differentiation\Nproblems, these dy dx's just
Dialogue: 0,0:06:48.32,0:06:50.57,Default,,0000,0000,0000,,don't, it's not some rule\Nthat you should memorize.
Dialogue: 0,0:06:50.57,0:06:52.89,Default,,0000,0000,0000,,They come out naturally\Nfrom the chain rule.
Dialogue: 0,0:06:52.89,0:06:56.93,Default,,0000,0000,0000,,So we solved da dx,\Nthat is equal to this
Dialogue: 0,0:06:56.93,0:06:59.23,Default,,0000,0000,0000,,expression right here.
Dialogue: 0,0:06:59.23,0:07:07.13,Default,,0000,0000,0000,,Let me write it, it's equal to\N1 over cosine squared of b.
Dialogue: 0,0:07:07.13,0:07:07.88,Default,,0000,0000,0000,,Well what's b?
Dialogue: 0,0:07:07.88,0:07:10.64,Default,,0000,0000,0000,,I wrote it's cos x over y.
Dialogue: 0,0:07:10.64,0:07:16.92,Default,,0000,0000,0000,,Cosine squared of x over y\Ntimes all of this stuff over
Dialogue: 0,0:07:16.92,0:07:19.84,Default,,0000,0000,0000,,here, times all of this mess.
Dialogue: 0,0:07:19.84,0:07:25.67,Default,,0000,0000,0000,,1 over y plus, or maybe I\Nshould say minus, minus -- if I
Dialogue: 0,0:07:25.67,0:07:32.49,Default,,0000,0000,0000,,just simplify this, this is x\Nover y squared times dy dx.
Dialogue: 0,0:07:32.49,0:07:36.66,Default,,0000,0000,0000,,
Dialogue: 0,0:07:36.66,0:07:39.00,Default,,0000,0000,0000,,Then that is equal to\Nthe right hand side.
Dialogue: 0,0:07:39.00,0:07:48.49,Default,,0000,0000,0000,,It is equal to 1 plus dy dx.
Dialogue: 0,0:07:48.49,0:07:51.42,Default,,0000,0000,0000,,And now all we have to\Ndo is solve for dy dx.
Dialogue: 0,0:07:51.42,0:07:53.99,Default,,0000,0000,0000,,So let me just review\Nhow we got here.
Dialogue: 0,0:07:53.99,0:07:56.30,Default,,0000,0000,0000,,I went through the chain rule\Nat every step of the way, but
Dialogue: 0,0:07:56.30,0:07:58.17,Default,,0000,0000,0000,,once you get the hang of it,\Nyou can literally just go
Dialogue: 0,0:07:58.17,0:07:59.36,Default,,0000,0000,0000,,straight down this way.
Dialogue: 0,0:07:59.36,0:08:01.38,Default,,0000,0000,0000,,The way you think about\Nit is -- the right hand
Dialogue: 0,0:08:01.38,0:08:02.03,Default,,0000,0000,0000,,side I think you get it.
Dialogue: 0,0:08:02.03,0:08:04.38,Default,,0000,0000,0000,,The derivative of x is 1, the\Nderivative of y with respect
Dialogue: 0,0:08:04.38,0:08:06.56,Default,,0000,0000,0000,,to x, well that's just dy dx.
Dialogue: 0,0:08:06.56,0:08:09.01,Default,,0000,0000,0000,,But the left hand side, you\Ntake the derivative of
Dialogue: 0,0:08:09.01,0:08:11.63,Default,,0000,0000,0000,,the whole thing with\Nrespect to x over y.
Dialogue: 0,0:08:11.63,0:08:14.10,Default,,0000,0000,0000,,So that's just the derivative\Nof tangent is 1 over
Dialogue: 0,0:08:14.10,0:08:15.02,Default,,0000,0000,0000,,cosine squared.
Dialogue: 0,0:08:15.02,0:08:18.62,Default,,0000,0000,0000,,So it's 1 over cosine squared\Nof x over y, and you multiply
Dialogue: 0,0:08:18.62,0:08:23.53,Default,,0000,0000,0000,,that times the derivative of\Nx over y with respect to x.
Dialogue: 0,0:08:23.53,0:08:26.77,Default,,0000,0000,0000,,And the derivative of x over\Ny with respect to x is the
Dialogue: 0,0:08:26.77,0:08:28.97,Default,,0000,0000,0000,,derivative of-- and it gets\Ncomplicated, that's why it's
Dialogue: 0,0:08:28.97,0:08:31.59,Default,,0000,0000,0000,,good to do it on the side here\N-- but it's the derivative of
Dialogue: 0,0:08:31.59,0:08:34.15,Default,,0000,0000,0000,,x, which is 1 times 1 over y.
Dialogue: 0,0:08:34.15,0:08:39.68,Default,,0000,0000,0000,,Which is that term plus the\Nderivative of 1 over Y with
Dialogue: 0,0:08:39.68,0:08:44.20,Default,,0000,0000,0000,,respect to X, which is minus\N1 over y squared dy dx, from
Dialogue: 0,0:08:44.20,0:08:46.62,Default,,0000,0000,0000,,the chain rule, times dx.
Dialogue: 0,0:08:46.62,0:08:48.14,Default,,0000,0000,0000,,That's why it was good to do\Nover to the side so we don't
Dialogue: 0,0:08:48.14,0:08:49.36,Default,,0000,0000,0000,,make a careless mistake.
Dialogue: 0,0:08:49.36,0:08:51.41,Default,,0000,0000,0000,,But once you get used to it you\Ncould actually do that in your
Dialogue: 0,0:08:51.41,0:08:53.98,Default,,0000,0000,0000,,head, and of course, that\Nequals the right hand side.
Dialogue: 0,0:08:53.98,0:08:56.52,Default,,0000,0000,0000,,So from here on out it's\Njust pure algebra.
Dialogue: 0,0:08:56.52,0:08:59.14,Default,,0000,0000,0000,,Just to solve for our dy dx.
Dialogue: 0,0:08:59.14,0:09:01.49,Default,,0000,0000,0000,,So a good place to start is to\Nmultiply both sides of this
Dialogue: 0,0:09:01.49,0:09:04.91,Default,,0000,0000,0000,,equation times cosine\Nsquared of x over y.
Dialogue: 0,0:09:04.91,0:09:07.42,Default,,0000,0000,0000,,So obviously, that'll\Nturn to 1 on this side.
Dialogue: 0,0:09:07.42,0:09:14.97,Default,,0000,0000,0000,,And the left hand side will be\N1 over y minus x over y squared
Dialogue: 0,0:09:14.97,0:09:23.69,Default,,0000,0000,0000,,dy dx is equal to -- I have to\Nmultiply both side of the
Dialogue: 0,0:09:23.69,0:09:26.73,Default,,0000,0000,0000,,equation times this denominator\Nright here -- is equal to
Dialogue: 0,0:09:26.73,0:09:32.53,Default,,0000,0000,0000,,cosine squared of x over y plus\Ncosine squared of
Dialogue: 0,0:09:32.53,0:09:35.19,Default,,0000,0000,0000,,x over y dy dx.
Dialogue: 0,0:09:35.19,0:09:39.42,Default,,0000,0000,0000,,
Dialogue: 0,0:09:39.42,0:09:40.19,Default,,0000,0000,0000,,Now what can we do.
Dialogue: 0,0:09:40.19,0:09:44.21,Default,,0000,0000,0000,,We can subtract this cosine\Nsquared of x over y from both
Dialogue: 0,0:09:44.21,0:09:52.11,Default,,0000,0000,0000,,sides of the equation, and we\Nget 1 over y minus cosine
Dialogue: 0,0:09:52.11,0:09:53.71,Default,,0000,0000,0000,,squared of x over y.
Dialogue: 0,0:09:53.71,0:09:55.78,Default,,0000,0000,0000,,All I did is I subtracted\Nthis from both sides of the
Dialogue: 0,0:09:55.78,0:09:57.59,Default,,0000,0000,0000,,equation, so essentially I\Nmoved it over to the
Dialogue: 0,0:09:57.59,0:09:59.04,Default,,0000,0000,0000,,left hand side.
Dialogue: 0,0:09:59.04,0:10:01.04,Default,,0000,0000,0000,,What I'm trying to do is I'm\Ngoing to try to separate the
Dialogue: 0,0:10:01.04,0:10:04.81,Default,,0000,0000,0000,,non dy dx terms from\Nthe dy dx terms.
Dialogue: 0,0:10:04.81,0:10:06.75,Default,,0000,0000,0000,,So I want to bring this\Ndy dx term over to
Dialogue: 0,0:10:06.75,0:10:07.95,Default,,0000,0000,0000,,the right hand side.
Dialogue: 0,0:10:07.95,0:10:11.55,Default,,0000,0000,0000,,So let me add x over y\Nsquared dy dx to both sides.
Dialogue: 0,0:10:11.55,0:10:17.26,Default,,0000,0000,0000,,So then that is equal to x over\Ny -- let me write that in the
Dialogue: 0,0:10:17.26,0:10:21.07,Default,,0000,0000,0000,,color that I originally wrote\Nit in, a slightly
Dialogue: 0,0:10:21.07,0:10:21.47,Default,,0000,0000,0000,,different color.
Dialogue: 0,0:10:21.47,0:10:27.11,Default,,0000,0000,0000,,So that is x over y squared --\NI'll write the dy dx in orange.
Dialogue: 0,0:10:27.11,0:10:34.12,Default,,0000,0000,0000,,dy dx, and then you have this\Nterm, plus cosine squared
Dialogue: 0,0:10:34.12,0:10:36.88,Default,,0000,0000,0000,,of x over y dy dx.
Dialogue: 0,0:10:36.88,0:10:40.95,Default,,0000,0000,0000,,
Dialogue: 0,0:10:40.95,0:10:43.00,Default,,0000,0000,0000,,I think we're in\Nthe home stretch.
Dialogue: 0,0:10:43.00,0:10:46.41,Default,,0000,0000,0000,,Let's factor this dy dx out\Nfrom the right hand side.
Dialogue: 0,0:10:46.41,0:10:56.77,Default,,0000,0000,0000,,So this is equal to dy dx\Ntimes x over y squared plus
Dialogue: 0,0:10:56.77,0:11:01.22,Default,,0000,0000,0000,,cosine squared of x over y.
Dialogue: 0,0:11:01.22,0:11:04.18,Default,,0000,0000,0000,,And that is equal to this thing\Nover here, it's equal to
Dialogue: 0,0:11:04.18,0:11:09.25,Default,,0000,0000,0000,,1 over y minus cosine\Nsquared of x over y.
Dialogue: 0,0:11:09.25,0:11:12.24,Default,,0000,0000,0000,,Now to solve for dy dx, we just\Nhave to divide both sides of
Dialogue: 0,0:11:12.24,0:11:15.45,Default,,0000,0000,0000,,this equation by this\Nexpression right here.
Dialogue: 0,0:11:15.45,0:11:16.90,Default,,0000,0000,0000,,And then what do we get?
Dialogue: 0,0:11:16.90,0:11:21.97,Default,,0000,0000,0000,,We get, if we just divide both\Nsides by that, we get 1 over y
Dialogue: 0,0:11:21.97,0:11:27.21,Default,,0000,0000,0000,,minus cosine squared of x over\Ny divided by this whole
Dialogue: 0,0:11:27.21,0:11:28.72,Default,,0000,0000,0000,,business right there.
Dialogue: 0,0:11:28.72,0:11:36.19,Default,,0000,0000,0000,,x over y squared plus cosine\Nsquared of x over y is
Dialogue: 0,0:11:36.19,0:11:42.15,Default,,0000,0000,0000,,equal to our dy dx.
Dialogue: 0,0:11:42.15,0:11:43.37,Default,,0000,0000,0000,,And then we're done.
Dialogue: 0,0:11:43.37,0:11:46.46,Default,,0000,0000,0000,,We just applied the chain rule\Nmultiple times and we were able
Dialogue: 0,0:11:46.46,0:11:50.60,Default,,0000,0000,0000,,to implicitly differentiate\Ntangent of y over x is
Dialogue: 0,0:11:50.60,0:11:51.60,Default,,0000,0000,0000,,equal to x plus y.
Dialogue: 0,0:11:51.60,0:11:55.98,Default,,0000,0000,0000,,The hard part really is getting\Nto this step right here.
Dialogue: 0,0:11:55.98,0:11:59.47,Default,,0000,0000,0000,,After this step it's literally\Njust pure algebra just to solve
Dialogue: 0,0:11:59.47,0:12:04.73,Default,,0000,0000,0000,,for the dy dx's, and then you\Nget that answer right there.
Dialogue: 0,0:12:04.73,0:12:07.38,Default,,0000,0000,0000,,Anyway, hopefully you\Nfound that useful.