WEBVTT 1 00:00:00.550 --> 00:00:02.060 When we wanna take derivatives, 2 00:00:02.060 --> 00:00:04.340 we've looked at a couple different ways of doing that. 3 00:00:04.340 --> 00:00:05.940 Most commonly, people have seen 4 00:00:05.940 --> 00:00:07.820 the Power Rule at this point. 5 00:00:07.820 --> 00:00:09.950 What we're gonna see though, in the next few slides 6 00:00:09.950 --> 00:00:12.400 is that there are other techniques for taking derivatives, 7 00:00:12.400 --> 00:00:14.710 and these get into some pretty complicated rules 8 00:00:14.710 --> 00:00:16.460 that many people memorize as they go 9 00:00:16.460 --> 00:00:18.530 into higher calculus studies. 10 00:00:18.530 --> 00:00:19.950 So, the first thing is it says 11 00:00:19.950 --> 00:00:21.540 we're gonna look at the Chain Rule. 12 00:00:21.540 --> 00:00:23.090 There are many rules for differentiation, 13 00:00:23.090 --> 00:00:24.540 depending on the function. 14 00:00:24.540 --> 00:00:26.100 One method, the Chain Rule, 15 00:00:26.100 --> 00:00:28.460 applies to a function within a function, 16 00:00:28.460 --> 00:00:30.260 or a composition of functions. 17 00:00:30.260 --> 00:00:33.350 So, if I have f of x is equal to f of g of x, 18 00:00:33.350 --> 00:00:36.910 then if I wanna use the Chain Rule, here's how it works: 19 00:00:36.910 --> 00:00:41.790 d y d x is equal to d y over d u times d u over d x. 20 00:00:41.790 --> 00:00:44.150 And here's an example: let's say my function is 21 00:00:44.150 --> 00:00:47.030 one plus x squared quantity to the fifth, 22 00:00:47.030 --> 00:00:48.270 so I have two functions going on. 23 00:00:48.270 --> 00:00:50.700 My first function is x to the fifth, 24 00:00:50.700 --> 00:00:52.370 and my second function that's going on 25 00:00:52.370 --> 00:00:54.010 is one plus x squared. 26 00:00:54.010 --> 00:00:55.590 That's a composition of functions, 27 00:00:55.590 --> 00:00:57.540 it's where one function is the input 28 00:00:57.540 --> 00:00:58.790 into the other function. 29 00:00:58.790 --> 00:01:01.270 So to calculate a derivative using the Chain Rule, 30 00:01:01.270 --> 00:01:04.230 what I'm gonna do is a substitution using the letter u. 31 00:01:04.230 --> 00:01:06.670 I'm gonna let u represent my inside function, 32 00:01:06.670 --> 00:01:09.100 or let u be one plus x squared, 33 00:01:09.100 --> 00:01:12.210 and then y would be equal to u to the fifth. 34 00:01:12.210 --> 00:01:14.020 Pause real quick and make sure that makes sense to you 35 00:01:14.020 --> 00:01:17.170 because it's called a u or dummy substitution, 36 00:01:17.170 --> 00:01:19.900 that inside now instead of being called one plus x squared, 37 00:01:19.900 --> 00:01:21.670 is just called the letter u. 38 00:01:21.670 --> 00:01:23.080 So when I go to do the derivative, 39 00:01:23.080 --> 00:01:25.850 d y d x would be the derivative of u squared 40 00:01:25.850 --> 00:01:30.180 over d u times the derivative of one plus x squared d x. 41 00:01:30.180 --> 00:01:32.650 And the derivative of u to the fifth 42 00:01:32.650 --> 00:01:34.760 would be five u to the fourth 43 00:01:34.760 --> 00:01:37.200 multiplied by the derivative of u. 44 00:01:37.200 --> 00:01:38.430 Derivative of u there's back to 45 00:01:38.430 --> 00:01:42.673 x is instead of x squared, it's gonna be two x. 46 00:01:43.750 --> 00:01:46.010 And then, if I simplify that, I'll have 47 00:01:46.010 --> 00:01:48.450 ten x u to the fourth, 48 00:01:48.450 --> 00:01:49.810 now don't leave your answer there, 49 00:01:49.810 --> 00:01:51.670 it's a common mistake, is that students 50 00:01:51.670 --> 00:01:53.100 will leave u in their answer. 51 00:01:53.100 --> 00:01:54.880 They did the derivative, they think they're done, 52 00:01:54.880 --> 00:01:57.330 but keep in mind u is something that I introduced, 53 00:01:57.330 --> 00:01:58.780 it wasn't in the original problem. 54 00:01:58.780 --> 00:02:01.320 So, to do this correctly, I need to get rid of that u. 55 00:02:01.320 --> 00:02:04.220 Or I'll go back and instead of having ten x u 56 00:02:04.220 --> 00:02:06.000 to the fourth, more appropriately, 57 00:02:06.000 --> 00:02:08.040 it would be ten x times the quantity 58 00:02:08.040 --> 00:02:10.213 one plus x squared to the fourth. 59 00:02:12.600 --> 00:02:13.580 There are similar rules that 60 00:02:13.580 --> 00:02:15.180 happen with products and quotients. 61 00:02:15.180 --> 00:02:18.040 On your test, you're gonna wanna look out for trick answers. 62 00:02:18.040 --> 00:02:20.070 If you're asked to take a derivative 63 00:02:20.070 --> 00:02:22.170 and you notice it's a compositional functions 64 00:02:22.170 --> 00:02:24.980 or a product of functions or a quotient of functions, 65 00:02:24.980 --> 00:02:26.600 and if you don't have these rules memorized, 66 00:02:26.600 --> 00:02:28.750 you're just gonna wanna know that there is a rule, 67 00:02:28.750 --> 00:02:30.170 and there's probably gonna be a trick answer 68 00:02:30.170 --> 00:02:31.550 presented for you. 69 00:02:31.550 --> 00:02:33.400 If I'm asked to do the derivative of a product, 70 00:02:33.400 --> 00:02:35.300 the product of f times g, 71 00:02:35.300 --> 00:02:38.120 I would do f times the derivative of g 72 00:02:38.120 --> 00:02:40.210 plus g times the derivative of f. 73 00:02:40.210 --> 00:02:42.520 It's like the first function as it is 74 00:02:42.520 --> 00:02:44.290 times the derivative of the second 75 00:02:44.290 --> 00:02:46.690 plus the second function as it is 76 00:02:46.690 --> 00:02:48.273 times derivative of the first. 77 00:02:49.150 --> 00:02:50.300 Or if I wanna do a quotient, 78 00:02:50.300 --> 00:02:53.400 let's say I wanna do the quotient of f divided by g, 79 00:02:53.400 --> 00:02:55.720 I would do g times the derivative of f 80 00:02:55.720 --> 00:02:57.730 minus f times the derivative of g 81 00:02:57.730 --> 00:03:00.000 divided by g squared. 82 00:03:00.000 --> 00:03:02.710 So again, it's not just derivative of the top, 83 00:03:02.710 --> 00:03:04.300 derivative of the bottom, and dividing them, 84 00:03:04.300 --> 00:03:05.740 that would be a trick answer. 85 00:03:05.740 --> 00:03:07.030 The correct way to do a derivative 86 00:03:07.030 --> 00:03:08.823 of a quotient is on your screen. 87 00:03:10.510 --> 00:03:14.290 Another type of standard technique for taking derivatives 88 00:03:14.290 --> 00:03:16.040 is if you have a logarithmic function. 89 00:03:16.040 --> 00:03:17.730 Here we're looking at the natural log, 90 00:03:17.730 --> 00:03:19.300 and then also we'll see what happens 91 00:03:19.300 --> 00:03:21.620 if you have an exponential function. 92 00:03:21.620 --> 00:03:23.040 The derivative of the natural log 93 00:03:23.040 --> 00:03:26.600 of absolute value of x is one over x, 94 00:03:26.600 --> 00:03:28.740 and so looking ahead, if you're working 95 00:03:28.740 --> 00:03:30.260 with an integration problem, 96 00:03:30.260 --> 00:03:32.340 which of course is where you're doing this backwards, 97 00:03:32.340 --> 00:03:34.390 if you see one over x, you know 98 00:03:34.390 --> 00:03:35.990 the anti-derivative is the natural log 99 00:03:35.990 --> 00:03:38.100 of absolute value of x. 100 00:03:38.100 --> 00:03:40.140 Back to derivatives, the derivative of e to the x 101 00:03:40.140 --> 00:03:41.450 is equal to e to the x. 102 00:03:41.450 --> 00:03:42.680 That's kinda cool. 103 00:03:42.680 --> 00:03:45.120 And the last one, the derivative of a to the x 104 00:03:45.120 --> 00:03:48.610 is equal to a to the x times the natural log of a. 105 00:03:48.610 --> 00:03:50.920 Again, these are rules that people might memorize 106 00:03:50.920 --> 00:03:53.710 as they go into higher level calculus studies. 107 00:03:53.710 --> 00:03:54.910 For you, you're gonna wanna be aware 108 00:03:54.910 --> 00:03:57.140 that there are some kind of trick answers. 109 00:03:57.140 --> 00:03:59.220 Like, if you're given that quotient 110 00:03:59.220 --> 00:04:01.060 a trick answer would be just the quotient 111 00:04:01.060 --> 00:04:03.420 of a derivatives which is not correct. 112 00:04:03.420 --> 00:04:06.550 Now moving on, there's trigonometric derivatives as well. 113 00:04:06.550 --> 00:04:07.990 So, on your screen you'll see that 114 00:04:07.990 --> 00:04:09.550 derivative of sine is cosine, 115 00:04:09.550 --> 00:04:11.810 derivative of tangent is secant squared, 116 00:04:11.810 --> 00:04:14.350 derivative of secant is secant x tangent x, 117 00:04:14.350 --> 00:04:16.730 the derivative of sine negative one of x 118 00:04:16.730 --> 00:04:19.483 is one over the square root of one minus x squared. 119 00:04:20.400 --> 00:04:22.810 Down the right hand column, the derivative of cosine x 120 00:04:22.810 --> 00:04:24.200 is negative sine x, 121 00:04:24.200 --> 00:04:27.240 derivative of cotangent x is negative cosecant x, 122 00:04:27.240 --> 00:04:28.390 derivative of cosecant x 123 00:04:28.390 --> 00:04:30.890 is negative cosecant x cotangent x 124 00:04:30.890 --> 00:04:32.920 and the derivative of tangent negative one 125 00:04:32.920 --> 00:04:35.490 would be one over one plus x squared. 126 00:04:35.490 --> 00:04:37.382 So again, these are rules that you could memorize, 127 00:04:37.382 --> 00:04:39.820 or just be aware that there are some 128 00:04:39.820 --> 00:04:43.313 unique properties for higher level derivatives.