1 00:00:00,000 --> 00:00:07,387 I want to differentiate tangent theta. How am I going to do this? 2 00:00:07,387 --> 00:00:18,415 Well, hopefully you remember that tangent theta is sine theta over cosine theta. 3 00:00:18,415 --> 00:00:22,396 Why is this? If you think back to the business about 4 00:00:22,396 --> 00:00:27,856 the right triangles, right, sine is this opposite side that I'm calling y over the 5 00:00:27,856 --> 00:00:31,226 hypotenuse. And cosine is this adjacent side whose 6 00:00:31,226 --> 00:00:35,675 length I'll call x over the hypotenuse. I'll call this angle theta. 7 00:00:35,675 --> 00:00:40,933 Then sine theta is y over r, and cosine theta is x over r, so this fraction can 8 00:00:40,933 --> 00:00:44,978 be simplified to y over x. That's the opposite side over the 9 00:00:44,978 --> 00:00:47,540 adjacent side. That's tangent of theta. 10 00:00:47,540 --> 00:00:51,200 So good. I can express Tangent theta as sin theta 11 00:00:51,200 --> 00:00:56,804 over cosine theta, how does that help? Well, I know how to differentiate sine 12 00:00:56,804 --> 00:01:02,482 and cosine, and by writing tangent this way, tangents now a quotient, so I can 13 00:01:02,482 --> 00:01:04,351 use the. Quotient rule. 14 00:01:04,351 --> 00:01:11,527 All right, so I'll use the quotient rule, and the derivative of tangent theta is 15 00:01:11,527 --> 00:01:19,067 the derivative of the numerator, which is cosine, times the der, denominator, which 16 00:01:19,067 --> 00:01:25,879 is just cosine, minus the derivative of the denominator, which is minus sine, 17 00:01:25,879 --> 00:01:31,965 times the numerator, which is sine. And this whole thing is over the 18 00:01:31,965 --> 00:01:35,690 denominator squared, cosine squared theta. 19 00:01:35,690 --> 00:01:40,470 Now, is this very helpful? Well, I've got cosine times cosine, so I 20 00:01:40,470 --> 00:01:46,221 can write that as cosine squared theta. And here I've got minus negative sine 21 00:01:46,221 --> 00:01:51,375 theta times sine theta, so that ends up being plus sine squared theta. 22 00:01:51,375 --> 00:01:56,840 And the whole thing is still being divided by cosine squared theta. 23 00:01:56,840 --> 00:02:02,140 Can I simplify that at all? Well cosine squared plus sine squared, 24 00:02:02,140 --> 00:02:06,055 that's the Pythagorean identity. That's just one. 25 00:02:06,055 --> 00:02:12,335 So I can replace the numerator with just one, still over cosine squared theta. 26 00:02:12,335 --> 00:02:18,859 And one over cosine squared theta, well, one over cosine is called secant, so this 27 00:02:18,859 --> 00:02:22,040 is really secant squared theta. . 28 00:02:22,040 --> 00:02:27,486 So what all this shows is the derivative of tangent theta is second squared theta. 29 00:02:27,486 --> 00:02:32,402 A moment ago we did a calculation using the quotient rule to see that the 30 00:02:32,402 --> 00:02:35,856 derivative of tangent theta is second squared theta. 31 00:02:35,856 --> 00:02:41,302 And now I want to see how this plays out in some concrete example to get a, a real 32 00:02:41,302 --> 00:02:44,225 sense as to why a formula like this is true. 33 00:02:44,225 --> 00:02:48,609 So here's a couple triangles. They're both right triangles and the 34 00:02:48,609 --> 00:02:53,897 length of their hypotenuse is the same. This angle I am calling alpha, let's this 35 00:02:53,897 --> 00:02:58,200 be a little bit less than 45 degree. And this angle I am calling beta, and 36 00:02:58,200 --> 00:03:02,150 it's more than 45 degrees. Now what you can say about the Secant of 37 00:03:02,150 --> 00:03:06,545 alpha and the Secant of beta? You know, the Secant of alpha is 38 00:03:06,545 --> 00:03:10,063 definitely bigger than one. I mean, what's the Secant? 39 00:03:10,063 --> 00:03:13,853 It's this hypotenuse length divided by this length here. 40 00:03:13,853 --> 00:03:17,780 And that's bigger than one. How does it compare to beta? 41 00:03:17,780 --> 00:03:22,745 Well, the secant of beta is quite a bit bigger than the secant of alpha, and why 42 00:03:22,745 --> 00:03:25,700 is that? Well, the secant of beta is this length 43 00:03:25,700 --> 00:03:30,414 here, the hypotenuse, divided by this width, but this triangle is quite a bit 44 00:03:30,414 --> 00:03:34,751 narrower than this triangle. So the secant has the same numerator, the 45 00:03:34,751 --> 00:03:39,465 hypotenuse, the same length, but the denominator here is quite a bit smaller, 46 00:03:39,465 --> 00:03:44,305 and if the denominator's a lot smaller, then the ratio, which is the secant, is 47 00:03:44,305 --> 00:03:48,031 quite a bit bigger. Some of the secan of beta is bigger than 48 00:03:48,031 --> 00:03:51,798 the secan of alpha, and the secan of alpha is bigger than one. 49 00:03:51,798 --> 00:03:55,750 And that means that secan squared alpha is also bigger than one. 50 00:03:55,750 --> 00:03:59,208 And secan squared alpha is less than secan squared beta. 51 00:03:59,208 --> 00:04:04,025 And the significance of that is right here, the derivative of tangent theta is 52 00:04:04,025 --> 00:04:06,584 secan squared theta. So what does this mean? 53 00:04:06,584 --> 00:04:10,268 Well, this, this is telling me how wiggling theta affects tangent. 54 00:04:10,268 --> 00:04:13,261 It affects it like a factor of secant squared theta. 55 00:04:13,261 --> 00:04:17,924 So in this example, where secant squared beta is a lot bigger than secant squared 56 00:04:17,924 --> 00:04:22,529 alpha, the effect of wiggling beta on the tangent of beta should be a lot larger 57 00:04:22,529 --> 00:04:25,868 than the effect of wiggling alpha on the tangent of alpha. 58 00:04:25,868 --> 00:04:29,207 And you can see that. Let me draw a triangle, where I've 59 00:04:29,207 --> 00:04:33,410 wiggled the angle alpha up a little bit. I've made it a little bit bigger. 60 00:04:33,410 --> 00:04:36,216 But I'm going to make the same hypotenuse. 61 00:04:36,216 --> 00:04:41,360 All right, so this hypotenuse length is the same as this length, but I've made 62 00:04:41,360 --> 00:04:45,971 the angle a little bit bigger. And how is the tangent of the slightly 63 00:04:45,971 --> 00:04:49,044 larger alpha compared to the tangent of alpha. 64 00:04:49,044 --> 00:04:54,323 Well, the tangent of the slightly larger alpha is bigger than tangent alpha, but 65 00:04:54,323 --> 00:04:58,532 not by all that much. Now compare that to when I wiggle beta up 66 00:04:58,532 --> 00:05:02,880 by the same amount. I make beta a little bit bigger. 67 00:05:02,880 --> 00:05:05,664 Right. So I make beta a little bit bigger by the 68 00:05:05,664 --> 00:05:10,072 same amount that I made alpha larger. And I think about how that affects the 69 00:05:10,072 --> 00:05:12,994 tangent of beta. The tangent of beta is this height 70 00:05:12,994 --> 00:05:17,096 divided by, divided by this width. And you can think about it, I mean this, 71 00:05:17,096 --> 00:05:19,775 this height maybe isn't increasing a whole lot. 72 00:05:19,775 --> 00:05:23,364 But the width of this triangle is getting quite a bit narrower. 73 00:05:23,364 --> 00:05:27,638 And because it's the ratio of that height to that width the tangent of the 74 00:05:27,638 --> 00:05:31,684 perturbed value of beta is quite a bit larger than the tangent of beta. 75 00:05:31,684 --> 00:05:36,299 And you can, you know it's reflected in the fact the secant squared beta is a lot 76 00:05:36,299 --> 00:05:40,027 larger than secant squared alpha. So these, these kind of facts, right? 77 00:05:40,027 --> 00:05:44,039 The fact that the derivative of tangent is secant squared theta you, you can 78 00:05:44,039 --> 00:05:48,367 really get a sense for why these things might be true by thinking about triangles 79 00:05:48,367 --> 00:05:52,590 and how wiggling the angle will affect certain ratios of sides of the triangles. 80 00:05:52,590 --> 00:05:57,392 But, if this seems a little too abstract we can kind of pull back a little bit and 81 00:05:57,392 --> 00:06:01,686 do do a numeric example next. You know, and maybe the numerical example 82 00:06:01,686 --> 00:06:05,075 is sort of another way to see a formula like this in action. 83 00:06:05,075 --> 00:06:09,595 Let's do a numerical example to get a sense as to what you might do with the 84 00:06:09,595 --> 00:06:12,590 fact that the derivative of tangent is secen squared. 85 00:06:12,590 --> 00:06:18,306 Here's an example, let's try to approximate the value of tangent of 46 86 00:06:18,306 --> 00:06:21,483 degrees. Why is this an interesting example? 87 00:06:21,483 --> 00:06:25,593 Well, we know the tangent of 45 degrees exactly, all right. 88 00:06:25,593 --> 00:06:31,001 And figure that out by looking at a triangle, here's the angle, 45 degrees, a 89 00:06:31,001 --> 00:06:35,471 right triangle, because the an, angles add up to, 180 degrees. 90 00:06:35,471 --> 00:06:39,437 So it's 45 plus 90 plus what? Well, this must also be 45. 91 00:06:39,437 --> 00:06:43,980 It's an isosceles triangle now, so these two sides are the same. 92 00:06:43,980 --> 00:06:48,705 A tangent is the ratio of this side to this side because they are equal that 93 00:06:48,705 --> 00:06:51,774 ratio was one. So I know the tangent of 45 exactly. 94 00:06:51,774 --> 00:06:54,351 It's one. But I am trying to figure out an 95 00:06:54,351 --> 00:06:58,770 approximation for the tangent of 46 degrees, the derivative tells me how 96 00:06:58,770 --> 00:07:03,557 wrigly an input affects the output, so I can use this fact and the fact that I 97 00:07:03,557 --> 00:07:07,240 know the derivative to try to approximate the tangent of 46. 98 00:07:07,240 --> 00:07:13,566 In particular the tangent of 46 degrees is the tangent of 45 plus one degree. 99 00:07:13,566 --> 00:07:18,250 And here you can see how I am perturbing the input a bit. 100 00:07:18,250 --> 00:07:22,861 And this is exactly what the derivative would tell me something about. 101 00:07:22,861 --> 00:07:28,263 A little bit of bad news, the derivative of tangent is only secant squared if I do 102 00:07:28,263 --> 00:07:32,281 the measurement in radians. If I convert this to a problem in 103 00:07:32,281 --> 00:07:35,312 radians. With radians, says the tangent of pi/4, 104 00:07:35,312 --> 00:07:40,318 which is 45 degrees, plus with one degree in radians, is pi/180 radians, right? 105 00:07:40,318 --> 00:07:44,864 This is what I want to compute. I want to compute the tangent of pi/4 106 00:07:44,864 --> 00:07:48,421 plus pi/180. And I'll do that with approximation using 107 00:07:48,421 --> 00:07:52,506 the derivative. So according to the derivative this is 108 00:07:52,506 --> 00:07:58,700 about the tangent of pi over four, which is the tangent of 45 degrees, it's one. 109 00:07:58,700 --> 00:08:04,822 Plus the derivative of pi over four, which is secant squared pi over four. 110 00:08:04,822 --> 00:08:09,770 Times how much I wiggled the input by, which is pi over 180. 111 00:08:09,770 --> 00:08:13,005 I know the tangent of pi over four. It's one. 112 00:08:13,005 --> 00:08:16,094 What's the secant of pi over four squared? 113 00:08:16,094 --> 00:08:21,831 Well, if I pretend that these sides have length one, by the pythagorean theorem, 114 00:08:21,831 --> 00:08:25,214 this side must have length square root of two. 115 00:08:25,214 --> 00:08:31,171 The secant is hypotenuse over this width. So the secant of pi over four is square 116 00:08:31,171 --> 00:08:35,069 root of two. So secant squared pi over four is square 117 00:08:35,069 --> 00:08:39,380 root of two squared, which is two times pi over 180. 118 00:08:39,380 --> 00:08:46,118 Now if you know an approximation for pi, you can compute two times pi over 180 119 00:08:46,118 --> 00:08:49,746 plus one. And this is approximately 1.0349, 120 00:08:49,746 --> 00:08:57,684 and it keeps going Pi's irrational. And this is not so far off of the actual 121 00:08:57,684 --> 00:09:03,317 value. If we actually compute tangent of 46, the 122 00:09:03,317 --> 00:09:09,070 actual value is about 1.0355 and it keeps going. 123 00:09:09,070 --> 00:09:15,543 And· 1 0355. is awfully close to 1.0349. So we've 124 00:09:15,543 --> 00:09:24,173 successfully used the derivative to approximate the value of tangent 46 125 00:09:24,173 --> 00:09:25,132 degrees.