1 00:00:12,140 --> 00:00:16,414 Now that we've seen what a derivative means and what it has to do with rates of change, 2 00:00:16,414 --> 00:00:19,379 our next step is to learn how to actually compute these guys. 3 00:00:19,839 --> 00:00:22,939 As in, if I give you some kind of function with an explicit formula, 4 00:00:22,939 --> 00:00:26,039 you'd want to be able to find what the formula for its derivative is. 5 00:00:26,699 --> 00:00:30,341 Maybe it's obvious, but I think it's worth stating explicitly why this 6 00:00:30,341 --> 00:00:34,085 is an important thing to be able to do, why much of a calculus student's 7 00:00:34,085 --> 00:00:37,469 time ends up going towards grappling with derivatives of abstract 8 00:00:37,469 --> 00:00:41,060 functions rather than thinking about concrete rate of change problems. 9 00:00:42,219 --> 00:00:45,908 It's because a lot of real-world phenomena, the sort of things that 10 00:00:45,908 --> 00:00:49,543 we want to use calculus to analyze, are modeled using polynomials, 11 00:00:49,543 --> 00:00:53,559 trigonometric functions, exponentials, and other pure functions like that. 12 00:00:53,979 --> 00:00:58,284 So if you build up some fluency with the ideas of rates of change for those kinds of 13 00:00:58,284 --> 00:01:02,641 pure abstract functions, it gives you a language to more readily talk about the rates 14 00:01:02,642 --> 00:01:07,100 at which things change in concrete situations that you might be using calculus to model. 15 00:01:07,920 --> 00:01:12,141 But it is way too easy for this process to feel like just memorizing a list of rules, 16 00:01:12,141 --> 00:01:16,166 and if that happens, if you get that feeling, it's also easy to lose sight of the 17 00:01:16,165 --> 00:01:20,239 fact that derivatives are fundamentally about just looking at tiny changes to some 18 00:01:20,239 --> 00:01:24,019 quantity and how that relates to a resulting tiny change in another quantity. 19 00:01:24,780 --> 00:01:28,731 So in this video and in the next one, my aim is to show you how you can think 20 00:01:28,731 --> 00:01:31,671 about a few of these rules intuitively and geometrically, 21 00:01:31,671 --> 00:01:35,674 and I really want to encourage you to never forget that tiny nudges are at the 22 00:01:35,674 --> 00:01:36,739 heart of derivatives. 23 00:01:37,920 --> 00:01:41,280 Let's start with a simple function like f of x equals x squared. 24 00:01:41,620 --> 00:01:42,740 What if I asked you its derivative? 25 00:01:43,519 --> 00:01:47,037 That is, if you were to look at some value x, like x equals 2, 26 00:01:47,037 --> 00:01:50,332 and compare it to a value slightly bigger, just dx bigger, 27 00:01:50,332 --> 00:01:53,739 what's the corresponding change in the value of the function? 28 00:01:54,260 --> 00:01:54,700 dF. 29 00:01:55,620 --> 00:01:58,692 And in particular, what's dF divided by dx, the rate 30 00:01:58,692 --> 00:02:01,939 at which this function is changing per unit change in x. 31 00:02:03,159 --> 00:02:07,248 As a first step for intuition, we know that you can think of this ratio 32 00:02:07,248 --> 00:02:10,882 dF dx as the slope of a tangent line to the graph of x squared, 33 00:02:10,883 --> 00:02:15,200 and from that you can see that the slope generally increases as x increases. 34 00:02:15,840 --> 00:02:18,400 At zero, the tangent line is flat, and the slope is zero. 35 00:02:19,000 --> 00:02:21,259 At x equals 1, it's something a bit steeper. 36 00:02:22,599 --> 00:02:24,400 At x equals 2, it's steeper still. 37 00:02:25,120 --> 00:02:27,579 But looking at graphs isn't generally the best way 38 00:02:27,579 --> 00:02:30,040 to understand the precise formula for a derivative. 39 00:02:30,719 --> 00:02:34,933 For that, it's best to take a more literal look at what x squared actually means, 40 00:02:34,933 --> 00:02:38,840 and in this case let's go ahead and picture a square whose side length is x. 41 00:02:39,919 --> 00:02:43,119 If you increase x by some tiny nudge, some little dx, 42 00:02:43,120 --> 00:02:46,379 what's the resulting change in the area of that square? 43 00:02:47,719 --> 00:02:51,479 That slight change in area is what dF means in this context. 44 00:02:52,020 --> 00:02:55,777 It's the tiny increase to the value of f of x equals x squared, 45 00:02:55,776 --> 00:02:58,419 caused by increasing x by that tiny nudge dx. 46 00:02:59,360 --> 00:03:03,031 Now you can see that there's three new bits of area in this diagram, 47 00:03:03,031 --> 00:03:05,319 two thin rectangles and a minuscule square. 48 00:03:06,240 --> 00:03:10,106 The two thin rectangles each have side lengths of x and dx, 49 00:03:10,105 --> 00:03:13,780 so they account for 2 times x times dx units of new area. 50 00:03:18,240 --> 00:03:21,031 For example, let's say x was 3 and dx was 0.01, 51 00:03:21,031 --> 00:03:25,741 then that new area from these two thin rectangles would be 2 times 3 times 0.01, 52 00:03:25,741 --> 00:03:28,300 which is 0.06, about 6 times the size of dx. 53 00:03:29,699 --> 00:03:32,871 That little square there has an area of dx squared, 54 00:03:32,872 --> 00:03:36,960 but you should think of that as being really tiny, negligibly tiny. 55 00:03:37,699 --> 00:03:41,339 For example, if dx was 0.01, that would be only 0.0001, 56 00:03:41,340 --> 00:03:46,086 and keep in mind I'm drawing dx with a fair bit of width here just so we 57 00:03:46,086 --> 00:03:49,662 can actually see it, but always remember in principle, 58 00:03:49,662 --> 00:03:54,992 dx should be thought of as a truly tiny amount, and for those truly tiny amounts, 59 00:03:54,992 --> 00:03:59,674 a good rule of thumb is that you can ignore anything that includes a dx 60 00:03:59,674 --> 00:04:01,819 raised to a power greater than 1. 61 00:04:02,400 --> 00:04:05,879 That is, a tiny change squared is a negligible change. 62 00:04:07,500 --> 00:04:13,055 What this leaves us with is that dF is just some multiple of dx, and that multiple 2x, 63 00:04:13,055 --> 00:04:18,100 which you could also write as dF divided by dx, is the derivative of x squared. 64 00:04:19,040 --> 00:04:24,297 For example, if you were starting at x equals 3, then as you slightly increase x, 65 00:04:24,297 --> 00:04:29,683 the rate of change in the area per unit change in length added, dx squared over dx, 66 00:04:29,682 --> 00:04:34,427 would be 2 times 3, or 6, and if instead you were starting at x equals 5, 67 00:04:34,427 --> 00:04:38,980 then the rate of change would be 10 units of area per unit change in x. 68 00:04:41,220 --> 00:04:45,420 Let's go ahead and try a different simple function, f of x equals x cubed. 69 00:04:45,939 --> 00:04:48,038 This is going to be the geometric view of the stuff 70 00:04:48,038 --> 00:04:50,139 that I went through algebraically in the last video. 71 00:04:51,019 --> 00:04:55,640 What's nice here is that we can think of x cubed as the volume of an actual 72 00:04:55,641 --> 00:05:00,020 cube whose side lengths are x, and when you increase x by a tiny nudge, 73 00:05:00,019 --> 00:05:04,519 a tiny dx, the resulting increase in volume is what I have here in yellow. 74 00:05:04,860 --> 00:05:08,747 That represents all the volume in a cube with side lengths x plus dx 75 00:05:08,747 --> 00:05:12,579 that's not already in the original cube, the one with side length x. 76 00:05:13,579 --> 00:05:18,601 It's nice to think of this new volume as broken up into multiple components, 77 00:05:18,601 --> 00:05:22,385 but almost all of it comes from these three square faces, 78 00:05:22,386 --> 00:05:25,843 or said a little more precisely, as dx approaches 0, 79 00:05:25,843 --> 00:05:30,278 those three squares comprise a portion closer and closer to 100% of 80 00:05:30,278 --> 00:05:31,779 that new yellow volume. 81 00:05:33,839 --> 00:05:38,057 Each of those thin squares has a volume of x squared times dx, 82 00:05:38,057 --> 00:05:41,540 the area of the face times that little thickness dx. 83 00:05:42,220 --> 00:05:46,260 So in total this gives us 3x squared dx of volume change. 84 00:05:47,300 --> 00:05:51,062 And to be sure there are other slivers of volume here along the edges 85 00:05:51,062 --> 00:05:54,877 and that tiny one in the corner, but all of that volume is going to be 86 00:05:54,877 --> 00:05:58,639 proportional to dx squared, or dx cubed, so we can safely ignore them. 87 00:05:59,459 --> 00:06:03,466 Again this is ultimately because they're going to be divided by dx, 88 00:06:03,466 --> 00:06:07,117 and if there's still any dx remaining then those terms aren't 89 00:06:07,117 --> 00:06:10,300 going to survive the process of letting dx approach 0. 90 00:06:11,279 --> 00:06:14,434 What this means is that the derivative of x cubed, 91 00:06:14,435 --> 00:06:19,199 the rate at which x cubed changes per unit change of x, is 3 times x squared. 92 00:06:20,639 --> 00:06:25,185 What that means in terms of graphical intuition is that the slope of 93 00:06:25,185 --> 00:06:29,600 the graph of x cubed at every single point x is exactly 3x squared. 94 00:06:34,079 --> 00:06:38,786 And reasoning about that slope, it should make sense that this derivative is high on the 95 00:06:38,786 --> 00:06:42,805 left and then 0 at the origin and then high again as you move to the right, 96 00:06:42,805 --> 00:06:47,141 but just thinking in terms of the graph would never have landed us on the precise 97 00:06:47,141 --> 00:06:48,199 quantity 3x squared. 98 00:06:48,879 --> 00:06:53,060 For that we had to take a much more direct look at what x cubed actually means. 99 00:06:54,259 --> 00:06:57,560 Now in practice you wouldn't necessarily think of the square every 100 00:06:57,560 --> 00:06:59,926 time you're taking the derivative of x squared, 101 00:06:59,927 --> 00:07:03,278 nor would you necessarily think of this cube whenever you're taking 102 00:07:03,278 --> 00:07:04,560 the derivative of x cubed. 103 00:07:04,879 --> 00:07:08,399 Both of them fall under a pretty recognizable pattern for polynomial terms. 104 00:07:09,199 --> 00:07:13,341 The derivative of x to the fourth turns out to be 4x cubed, 105 00:07:13,341 --> 00:07:17,759 the derivative of x to the fifth is 5x to the fourth, and so on. 106 00:07:18,879 --> 00:07:22,791 Abstractly you'd write this as the derivative of x to 107 00:07:22,791 --> 00:07:26,559 the n for any power n is n times x to the n minus 1. 108 00:07:27,300 --> 00:07:30,560 This right here is what's known in the business as the power rule. 109 00:07:31,740 --> 00:07:35,913 In practice we all quickly just get jaded and think about this symbolically as 110 00:07:35,913 --> 00:07:39,769 the exponent hopping down in front, leaving behind one less than itself, 111 00:07:39,769 --> 00:07:44,259 rarely pausing to think about the geometric delights that underlie these derivatives. 112 00:07:45,240 --> 00:07:47,295 That's the kind of thing that happens when these tend 113 00:07:47,295 --> 00:07:49,199 to fall in the middle of much longer computations. 114 00:07:50,639 --> 00:07:53,327 But rather than tracking it all off to symbolic patterns, 115 00:07:53,327 --> 00:07:57,359 let's just take a moment and think about why this works for powers beyond just 2 and 3. 116 00:07:58,439 --> 00:08:02,773 When you nudge that input x, increasing it slightly to x plus dx, 117 00:08:02,773 --> 00:08:06,974 working out the exact value of that nudged output would involve 118 00:08:06,973 --> 00:08:10,519 multiplying together these n separate x plus dx terms. 119 00:08:11,339 --> 00:08:13,889 The full expansion would be really complicated, 120 00:08:13,889 --> 00:08:18,459 but part of the point of derivatives is that most of that complication can be ignored. 121 00:08:19,279 --> 00:08:22,019 The first term in your expansion is x to the n. 122 00:08:22,680 --> 00:08:25,584 This is analogous to the area of the original square, 123 00:08:25,584 --> 00:08:28,920 or the volume of the original cube from our previous examples. 124 00:08:30,819 --> 00:08:36,038 For the next terms in the expansion you can choose mostly x's with a single dx. 125 00:08:41,720 --> 00:08:46,803 Since there are n different parentheticals from which you could have chosen 126 00:08:46,803 --> 00:08:50,015 that single dx, this gives us n separate terms, 127 00:08:50,015 --> 00:08:53,159 all of which include n minus 1 x's times a dx, 128 00:08:53,159 --> 00:08:56,639 giving a value of x to the power n minus 1 times dx. 129 00:08:57,580 --> 00:09:02,820 This is analogous to how the majority of the new area in the square came from those 130 00:09:02,820 --> 00:09:07,997 two bars, each with area x times dx, or how the bulk of the new volume in the cube 131 00:09:07,996 --> 00:09:13,299 came from those three thin squares, each of which had a volume of x squared times dx. 132 00:09:14,539 --> 00:09:17,432 There will be many other terms of this expansion, 133 00:09:17,432 --> 00:09:21,250 but all of them are just going to be some multiple of dx squared, 134 00:09:21,250 --> 00:09:25,184 so we can safely ignore them, and what that means is that all but a 135 00:09:25,184 --> 00:09:29,349 negligible portion of the increase in the output comes from n copies of 136 00:09:29,350 --> 00:09:31,259 this x to the n minus 1 times dx. 137 00:09:31,940 --> 00:09:37,520 That's what it means for the derivative of x to the n to be n times x to the n minus 1. 138 00:09:38,960 --> 00:09:43,220 And even though, like I said in practice, you'll find yourself performing this 139 00:09:43,220 --> 00:09:47,911 derivative quickly and symbolically, imagining the exponent hopping down to the front, 140 00:09:47,910 --> 00:09:52,279 every now and then it's nice to just step back and remember why these rules work. 141 00:09:52,820 --> 00:09:56,879 Not just because it's pretty, and not just because it helps remind us that math 142 00:09:56,879 --> 00:10:00,331 actually makes sense and isn't just a pile of formulas to memorize, 143 00:10:00,331 --> 00:10:04,494 but because it flexes that very important muscle of thinking about derivatives in 144 00:10:04,494 --> 00:10:05,560 terms of tiny nudges. 145 00:10:07,500 --> 00:10:11,639 As another example, think of the function f of x equals 1 divided by x. 146 00:10:12,700 --> 00:10:16,738 Now on the hand you could just blindly try applying the power rule, 147 00:10:16,738 --> 00:10:20,540 since 1 divided by x is the same as writing x to the negative 1. 148 00:10:21,100 --> 00:10:24,432 That would involve letting the negative 1 hop down in front, 149 00:10:24,432 --> 00:10:27,439 leaving behind 1 less than itself, which is negative 2. 150 00:10:28,240 --> 00:10:31,443 But let's have some fun and see if we can reason about this geometrically, 151 00:10:31,443 --> 00:10:33,579 rather than just plugging it through some formula. 152 00:10:34,860 --> 00:10:40,180 The value 1 over x is asking what number multiplied by x equals 1. 153 00:10:40,960 --> 00:10:42,820 So here's how I'd like to visualize it. 154 00:10:42,820 --> 00:10:48,120 Imagine a little rectangular puddle of water sitting in two dimensions whose area is 1. 155 00:10:48,960 --> 00:10:53,766 And let's say that its width is x, which means that the height has to be 1 over x, 156 00:10:53,765 --> 00:10:55,620 since the total area of it is 1. 157 00:10:56,360 --> 00:11:01,039 So if x was stretched out to 2, then that height is forced down to 1 half. 158 00:11:01,779 --> 00:11:05,919 And if you increased x up to 3, then the other side has to be squished down to 1 third. 159 00:11:07,039 --> 00:11:10,679 This is a nice way to think about the graph of 1 over x, by the way. 160 00:11:11,279 --> 00:11:15,360 If you think of this width x of the puddle as being in the xy-plane, 161 00:11:15,360 --> 00:11:20,623 then that corresponding output 1 divided by x, the height of the graph above that point, 162 00:11:20,623 --> 00:11:24,940 is whatever the height of your puddle has to be to maintain an area of 1. 163 00:11:26,360 --> 00:11:29,358 So with this visual in mind, for the derivative, 164 00:11:29,357 --> 00:11:33,579 imagine nudging up that value of x by some tiny amount, some tiny dx. 165 00:11:34,580 --> 00:11:37,401 How must the height of this rectangle change so 166 00:11:37,400 --> 00:11:40,339 that the area of the puddle remains constant at 1? 167 00:11:41,340 --> 00:11:46,019 That is, increasing the width by dx adds some new area to the right here. 168 00:11:46,259 --> 00:11:50,355 So the puddle has to decrease in height by some d 1 over x, 169 00:11:50,355 --> 00:11:54,860 so that the area lost off of that top cancels out the area gained. 170 00:11:56,100 --> 00:11:59,259 You should think of that d 1 over x as being a negative amount, 171 00:11:59,259 --> 00:12:02,320 by the way, since it's decreasing the height of the rectangle. 172 00:12:03,539 --> 00:12:04,399 And you know what? 173 00:12:04,840 --> 00:12:07,027 I'm going to leave the last few steps here for you, 174 00:12:07,027 --> 00:12:09,720 for you to pause and ponder and work out an ultimate expression. 175 00:12:10,559 --> 00:12:14,120 And once you reason out what d of 1 over x divided by dx should be, 176 00:12:14,120 --> 00:12:17,838 I want you to compare it to what you would have gotten if you had just 177 00:12:17,839 --> 00:12:21,820 blindly applied the power rule, purely symbolically, to x to the negative 1. 178 00:12:23,980 --> 00:12:26,143 And while I'm encouraging you to pause and ponder, 179 00:12:26,143 --> 00:12:28,519 here's another fun challenge if you're feeling up to it. 180 00:12:29,059 --> 00:12:33,419 See if you can reason through what the derivative of the square root of x should be. 181 00:12:36,399 --> 00:12:40,052 To finish things off, I want to tackle one more type of function, 182 00:12:40,052 --> 00:12:44,259 trigonometric functions, and in particular let's focus on the sine function. 183 00:12:45,320 --> 00:12:48,230 So for this section I'm going to assume that you're already 184 00:12:48,230 --> 00:12:51,673 familiar with how to think about trig functions using the unit circle, 185 00:12:51,673 --> 00:12:54,100 the circle with a radius 1 centered at the origin. 186 00:12:55,240 --> 00:12:59,152 For a given value of theta, like say 0.8, you imagine yourself 187 00:12:59,152 --> 00:13:02,878 walking around the circle starting from the rightmost point 188 00:13:02,878 --> 00:13:06,480 until you've traversed that distance of 0.8 in arc length. 189 00:13:06,759 --> 00:13:11,717 This is the same thing as saying that the angle right here is exactly theta radians, 190 00:13:11,717 --> 00:13:13,759 since the circle has a radius of 1. 191 00:13:14,759 --> 00:13:20,013 Then what sine of theta means is the height of that point above the x-axis, 192 00:13:20,013 --> 00:13:24,507 and as your theta value increases and you walk around the circle 193 00:13:24,506 --> 00:13:28,239 your height bobs up and down between negative 1 and 1. 194 00:13:29,019 --> 00:13:33,615 So when you graph sine of theta versus theta you get this wave pattern, 195 00:13:33,615 --> 00:13:35,659 the quintessential wave pattern. 196 00:13:37,600 --> 00:13:40,311 And just from looking at this graph we can start to 197 00:13:40,311 --> 00:13:43,180 get a feel for the shape of the derivative of the sine. 198 00:13:44,019 --> 00:13:48,827 The slope at 0 is something positive since sine of theta is increasing there, 199 00:13:48,827 --> 00:13:54,191 and as we move to the right and sine of theta approaches its peak that slope goes down 200 00:13:54,191 --> 00:13:54,500 to 0. 201 00:13:55,720 --> 00:13:58,340 Then the slope is negative for a little while, 202 00:13:58,340 --> 00:14:03,080 while the sine is decreasing before coming back up to 0 as the sine graph levels out. 203 00:14:04,460 --> 00:14:07,432 And as you continue thinking this through and drawing it out, 204 00:14:07,432 --> 00:14:11,173 if you're familiar with the graph of trig functions you might guess that this 205 00:14:11,173 --> 00:14:13,668 derivative graph should be exactly cosine of theta, 206 00:14:13,668 --> 00:14:17,264 since all the peaks and valleys line up perfectly with where the peaks and 207 00:14:17,264 --> 00:14:19,279 valleys for the cosine function should be. 208 00:14:20,340 --> 00:14:23,910 And spoiler alert, the derivative is in fact the cosine of theta, 209 00:14:23,909 --> 00:14:27,860 but aren't you a little curious about why it's precisely cosine of theta? 210 00:14:28,240 --> 00:14:32,158 I mean you could have all sorts of functions with peaks and valleys at the same points 211 00:14:32,158 --> 00:14:34,365 that have roughly the same shape, but who knows, 212 00:14:34,365 --> 00:14:38,103 maybe the derivative of sine could have turned out to be some entirely new type of 213 00:14:38,102 --> 00:14:40,399 function that just happens to have a similar shape. 214 00:14:41,600 --> 00:14:44,831 Well just like the previous examples, a more exact understanding 215 00:14:44,831 --> 00:14:48,662 of the derivative requires looking at what the function actually represents, 216 00:14:48,662 --> 00:14:51,100 rather than looking at the graph of the function. 217 00:14:52,399 --> 00:14:54,995 So think back to that walk around the unit circle, 218 00:14:54,995 --> 00:14:58,966 having traversed an arc with length theta and thinking about sine of theta as 219 00:14:58,966 --> 00:15:00,240 the height of that point. 220 00:15:01,700 --> 00:15:06,247 Now zoom into that point on the circle and consider a slight nudge of d theta 221 00:15:06,246 --> 00:15:10,620 along their circumference, a tiny step in your walk around the unit circle. 222 00:15:11,480 --> 00:15:14,639 How much does that tiny step change the sine of theta? 223 00:15:15,440 --> 00:15:20,420 How much does this increase d theta of arc length increase the height above the x-axis? 224 00:15:21,639 --> 00:15:26,181 Well zoomed in close enough, the circle basically looks like a straight line in this 225 00:15:26,181 --> 00:15:30,777 neighborhood, so let's go ahead and think of this right triangle where the hypotenuse 226 00:15:30,777 --> 00:15:34,891 of that right triangle represents the nudge d theta along the circumference, 227 00:15:34,890 --> 00:15:39,539 and that left side here represents the change in height, the resulting d sine of theta. 228 00:15:40,139 --> 00:15:44,184 Now this tiny triangle is actually similar to this larger triangle here, 229 00:15:44,184 --> 00:15:48,840 with the defining angle theta and whose hypotenuse is the radius of the circle with 230 00:15:48,841 --> 00:15:49,340 length 1. 231 00:15:50,960 --> 00:15:55,940 Specifically this little angle right here is precisely equal to theta radians. 232 00:15:57,419 --> 00:16:00,519 Now think about what the derivative of sine is supposed to mean. 233 00:16:01,220 --> 00:16:05,585 It's the ratio between that d sine of theta, the tiny change to the height, 234 00:16:05,585 --> 00:16:09,320 divided by d theta, the tiny change to the input of the function. 235 00:16:10,519 --> 00:16:14,052 And from the picture we can see that that's the ratio between the 236 00:16:14,052 --> 00:16:17,960 length of the side adjacent to the angle theta divided by the hypotenuse. 237 00:16:18,799 --> 00:16:21,517 Well let's see, adjacent divided by hypotenuse, 238 00:16:21,518 --> 00:16:26,220 that's exactly what the cosine of theta means, that's the definition of the cosine. 239 00:16:27,539 --> 00:16:30,222 So this gives us two different really nice ways of 240 00:16:30,222 --> 00:16:32,959 thinking about how the derivative of sine is cosine. 241 00:16:33,139 --> 00:16:36,641 One of them is looking at the graph and getting a loose feel for the shape of 242 00:16:36,642 --> 00:16:40,280 things based on thinking about the slope of the sine graph at every single point. 243 00:16:41,100 --> 00:16:45,399 And the other is a more precise line of reasoning looking at the unit circle itself. 244 00:16:47,080 --> 00:16:49,276 For those of you that like to pause and ponder, 245 00:16:49,275 --> 00:16:52,846 see if you can try a similar line of reasoning to find what the derivative of 246 00:16:52,846 --> 00:16:54,220 the cosine of theta should be. 247 00:16:56,320 --> 00:16:59,496 In the next video I'll talk about how you can take derivatives 248 00:16:59,495 --> 00:17:02,470 of functions who combine simple functions like these ones, 249 00:17:02,470 --> 00:17:06,000 either as sums or products or function compositions, things like that. 250 00:17:06,559 --> 00:17:09,643 And similar to this video the goal is going to be to understand each one 251 00:17:09,643 --> 00:17:13,358 geometrically in a way that makes it intuitively reasonable and somewhat more memorable.