1 00:00:00,012 --> 00:00:08,308 [music] We've seen how to differentiate sine, and cosine and tangent, but how do I 2 00:00:08,308 --> 00:00:13,003 differentiate secant? So let's differentiate secant. 3 00:00:13,003 --> 00:00:18,301 And one way to get a handle on secant is to write secant as a composition of two 4 00:00:18,301 --> 00:00:22,627 different functions. If I set f of x equals 1 over x, and g of 5 00:00:22,627 --> 00:00:25,925 x equals cosine x. Then f of g of x is secant x. 6 00:00:25,925 --> 00:00:31,643 So if I want to differentiate secant, it'd be good enough to differentiate f of g of 7 00:00:31,643 --> 00:00:34,796 x. And I can do that differentiation problem 8 00:00:34,796 --> 00:00:38,859 by using the chain rule. Now I can separately calculate the 9 00:00:38,859 --> 00:00:42,431 derivative of 1 over, it's negative 1 over x squared. 10 00:00:42,431 --> 00:00:45,386 And the derivative of cosine is negative sine. 11 00:00:45,386 --> 00:00:50,408 So putting this together, the derivative of secant must be the derivative of f, 12 00:00:50,408 --> 00:00:54,408 which is minus 1 over its input squared at g of x and g is cosine. 13 00:00:54,409 --> 00:01:00,048 So, with negative 1 over cosine square that's f prime of g of x times the 14 00:01:00,048 --> 00:01:06,062 derivative of g which is minus sine. Now I can put this together, the minus 15 00:01:06,062 --> 00:01:10,879 signs cancel and I'm left with sine x over cosine squared x. 16 00:01:10,879 --> 00:01:17,992 But people don't usually write it this way I could instead write this, as sine x over 17 00:01:17,992 --> 00:01:23,942 cosine x times 1 over cosine x. In other words, I could write this as, 18 00:01:23,942 --> 00:01:31,286 what's this term, this is tangent and this is secant so I can write this as tangent x 19 00:01:31,286 --> 00:01:35,853 secant x. So the derivative of secant x is tangent x 20 00:01:35,853 --> 00:01:40,464 times secant x. We can play the same kind of game to 21 00:01:40,464 --> 00:01:46,470 differentiate cosecant. So what about the derivative of cosecant 22 00:01:46,470 --> 00:01:49,605 x? Well think about how I can rewrite 23 00:01:49,605 --> 00:01:54,894 cosecant, alright? Cosecant is the composition of the one 24 00:01:54,894 --> 00:02:00,175 over function and sine, right? Cosecant is one over sine. 25 00:02:00,175 --> 00:02:06,781 So this is the derivative of f of g of x, where f is 1 over and g is sine, and by 26 00:02:06,781 --> 00:02:13,452 the chain rule, that's the derivative of f at g times the derivative of g. 27 00:02:13,452 --> 00:02:17,574 And now I know what the derivatives of these functions are. 28 00:02:17,574 --> 00:02:23,496 The derivative of 1 over is minus 1 over x squared, and the derivative of g is cosine 29 00:02:23,496 --> 00:02:26,945 x. And consequently, the derivative of 30 00:02:26,945 --> 00:02:33,289 cosecant x must be minus 1 over, that's the derivative of f at g of x which is 31 00:02:33,289 --> 00:02:39,495 sine, so it's sine squared times the derivative of g, which is cosine x. 32 00:02:39,495 --> 00:02:46,321 And I can combine these to get minus cosine x over sine x times 1 over sine x. 33 00:02:46,321 --> 00:02:53,399 In other words, minus cosine over sine, that's minus cotangent x, and minus 1 over 34 00:02:53,399 --> 00:02:59,857 sine, well that's cosecant again. So the derivative of cosecant x is minus 35 00:02:59,857 --> 00:03:04,592 cotangent x times cosecant x. And we can complete the story by 36 00:03:04,592 --> 00:03:09,438 differentiating cotangent. Okay, so how do I differentiate cotangent 37 00:03:09,438 --> 00:03:12,073 x ? Well we actually have some choices. 38 00:03:12,073 --> 00:03:17,182 One way would be to write this as the derivative of cosine x over sine x, since 39 00:03:17,182 --> 00:03:21,288 cotangent is cosine over sine. And then about the quotient rule. 40 00:03:21,288 --> 00:03:26,745 Or, I could use the fact that cotangent is one over tangent, and then differentiate 41 00:03:26,745 --> 00:03:31,716 this using the chain rule. To differentiate 1 over something, that's 42 00:03:31,716 --> 00:03:37,501 negative 1 over the thing squared times the derivative of the inside function 43 00:03:37,501 --> 00:03:41,748 which is tangent x. And I know the derivative of tangent x 44 00:03:41,748 --> 00:03:47,711 that's secant squared, so this is negative 1 over tangent squared x times secant 45 00:03:47,711 --> 00:03:53,763 squared x and that means the derivative of cotangent is negative secant squared x 46 00:03:53,763 --> 00:03:55,408 over. Tangent squared x. 47 00:03:55,408 --> 00:03:58,112 But people don't usually write it like this. 48 00:03:58,112 --> 00:04:02,652 Instead, I could simplify this. Having a secant squared in the numerator 49 00:04:02,652 --> 00:04:07,602 is as good as having a cosine squared in the denominator, and having a tangent in 50 00:04:07,602 --> 00:04:12,402 the denominator, Is as good as having cosines, in the numerator, and sines in 51 00:04:12,402 --> 00:04:16,313 the denominator. So, a negative secant squared over tangent 52 00:04:16,313 --> 00:04:21,253 squared is the same as negative cosine squared over cosine squared times sine 53 00:04:21,253 --> 00:04:24,565 squared. Now, these cosines cancel And what I'm 54 00:04:24,565 --> 00:04:27,645 left with is negative 1 over sine squared x. 55 00:04:27,645 --> 00:04:31,001 But people don't even write it this way, right? 56 00:04:31,001 --> 00:04:36,852 1 over sine, well that's cosecant so this is actually negative cosecant squared x. 57 00:04:36,852 --> 00:04:42,318 Now that we've seen the derivatives of all six of our trig functions, how are we 58 00:04:42,318 --> 00:04:46,811 suppose to remember these derivatives? Well here's a table. 59 00:04:46,811 --> 00:04:50,836 Showing all the derivatives of these 6 trig functions. 60 00:04:50,836 --> 00:04:54,816 And you can see there's some real pattern to this table. 61 00:04:54,816 --> 00:05:00,042 The derivative of sine is cosine. The derivative of cosine has a negative 62 00:05:00,042 --> 00:05:03,477 sine. The derivative of tangent we saw is secant 63 00:05:03,477 --> 00:05:06,657 squared. And the derivative of cotangent is 64 00:05:06,657 --> 00:05:13,500 cosecant but with a negative sine. And the derivative of secant we just saw 65 00:05:13,500 --> 00:05:20,071 is secant tangent. And the derivative of cosecant is negative 66 00:05:20,071 --> 00:05:26,891 cosecant cotangent. So there's definitely some symmetry here, 67 00:05:26,891 --> 00:05:34,968 and you can exploit that symmetry to help you to remember these derivatives.