03 - Derivative formulas through geometry
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A few derivative formulas, such as the power rule and the derivative of sine, demonstrated with geometric intuition. Brought to you by you: http://3b1b.co/eoc3-thanksAnd by Brilliant: https://brilliant.org/3b1bGreat video by Think Twice showing this geometric view of the derivative of sin(x)
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Now that we've seen what a derivative means, and what it has to do with rates of change.
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Our next step is to learn how to actually compute these guys, as in if I give you some kind of function
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with an explicit formula you'd want to be able to find what the formula for its derivative is.
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Maybe it's obvious, but I think it's worth stating explicitly why this is an important thing
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to be able to do. Why much of a calculus students time ends up going towards grappling with
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derivatives of abstract functions rather than thinking about concrete rate of change problems,
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Is because a lot of real-world phenomena. The sort of things that we want to use calculus
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to analyze are modeled using polynomials, trigonometric functions, exponential's
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and other pure functions like that. So if you build up some fluency with the ideas
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of rates of change for those kinds of pure abstract functions. It gives you a language to more
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readily talk about the rates at which things change in concrete situations that you might be
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using calculus to model. But it is way too easy for this process to feel like just memorizing a list
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of rules. And if that happens, if you get that feeling it's all so easy to lose sight of the fact that
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derivatives are fundamentally about just looking at tiny changes to some quantity,
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and how that relates to a resulting tiny change in another quantity. So in this video
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and in the next one, my aim is to show you how you can think about a few of these rules intuitively
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and geometrically. And I really want to encourage you to never forget that tiny nudges
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are at the heart of derivatives. let's start with a simple function like f(x) = x2, what if I asked
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you it's derivative. That is if you were to look at some value x like x = 2 and compare it to a value
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slightly bigger, just dx bigger. What's the corresponding change in the value of the
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function df, and in particular what's df divided by dx - The rate at which this function is changing
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per unit change in x as a first step for intuition. We know that you can think of this ratio df/dx
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as the slope of a tangent line to the graph of x2, and from that you can see that the slope
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generally increases as x increases. At 0 the tangent line is flat so the slope is 0 at x = 1.
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That's something a bit steeper at x = 2, it's steeper still but looking at graphs isn't generally the
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best way to understand the precise formula for a derivative, for that it's best to take
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a more literal look at what x2 actually means. And in this case let's go ahead and picture a square
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whose side length is x if you increase x by some tiny nudge, some little dx.
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What's the resulting change in the area of that square, that slight change, in area
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is what df means in this context. It's the tiny increase to the value of f(x) = x2 caused by
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increasing x by that tiny nudge dx. Now you can see that there's three new bits of area in this diagram, two thin
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rectangles and a miniscule square. The two thin rectangles each have side lengths of x and dx
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so they account for two times x * dx units of new area. For example let's say x was 3 and dx was 0.01.
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Then that new area from these two thin rectangles would be 2 * 3 * 0.01 which is 0.06,
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about 6 times the size of dx. That little square there has an area of dx2, but you should
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think of that as being really tiny, negligibly tiny. For example, if dx was 0.01, that would be only 0.0001.
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And keep in mind, I'm drawing DX with a fair bit of width here just so we can actually
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see it, but always remember - in principle dx should be thought of as a truly tiny amount
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and for those truly tiny amounts. A good rule of thumb is that you can ignore anything that includes a dx
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raised to a power greater than 1. That is a tiny change squared is a negligible change what this leaves us with
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is that df is just some multiple of dx, and that multiple 2x which you could also write as df/dx
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is the derivative of x2. For example, if you were starting at x = 3, then as you slightly increase x
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the rate of change in the area per unit change in length added dx2/dx would be 2 * 3 or 6.
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And if instead you were starting at x = 5, the rate of change would be ten units of area per unit change in x.
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Let's go ahead and try a different simple function f(x) = x3. This is going to be the geometric view
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of the stuff that I went through algebraically in the last video. What's nice here is that we can think of x3
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as the volume of an actual cube, whose side lengths are x. And when you increase x by a tiny nudge, a tiny dx
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the resulting increase in volume is what I have here in yellow. That represents all the volume in a cube
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with side lengths x plus dx. That's not already in the original cube, the one with side length x.
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It's nice to think of this new volume as broken up into multiple components, but almost all of it comes
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from these three square faces, or set a little more precisely as dx approaches zero.
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Precisely as dx approaches zero, those three squares comprise a portion closer and closer to 100%
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of that new yellow volume. Each of those thin squares has a volume of x2 * dx,
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the area of the face times that little thickness dx. So in total this gives us 3x2 dx of volume change.
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And to be sure there are other slivers of volume here, along the edges, and that tiny one in the corner. But all
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of that volume is going to be proportional to dx2 or dx3 so we can safely ignore them.
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Again this is ultimately because they're going to be divided by dx, and if there's still any dx remaining
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then those terms aren't going to survive the process of letting dx approach 0. What this means is
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that the derivative of x3, the rate at which x3 changes per unit change of x is 3x2.
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What that means in terms of graphical intuition is that the slope of the graph of x3
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at every single point x is exactly 3x2. And reasoning about that slope, it should make sense
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that this derivative is high on the left, and then zero at the origin, and then high again as you move to the right.
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But just thinking in terms of the graph would never have landed us on the precise quantity 3x2.
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For that we had to take a much more direct look at what x3 actually means now in practice.
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You wouldn't necessarily think of the square every time you're taking the derivative of x2. Nor would
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you necessarily think of this cube whenever you're taking the derivative of x3, both of them fall under a pretty
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recognizable pattern for polynomial terms the derivative of x4 turns out to be 4x3. The derivative
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of x5 is 5x4. The derivative of x to the n for any power n is nx(n-1).
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This right here is what's known in the business as the power rule. In practice
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we all quickly just get jaded and think about this symbolically as the exponent
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hopping down in front leaving behind one less than itself. Rarely pausing to think about the
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geometric delights that underlie these derivatives. That's the kind of thing
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that happens when these tend to fall in the middle of much longer computations.
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But rather than tracking it all off to symbolic patterns let's just take a moment and think about why this works.
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For powers beyond just 2 and 3 when you nudge that input x, increasing it slightly to x + dx, working out the
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exact value of that nudged output would involve multiplying together these n separate x + dx terms.
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The full expansion would be really complicated but part of the point of derivatives is that most of that
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complication can be ignored. The first term in your expansion is xn, this is analogous to the
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area of the original square or the volume of the original cube from our previous examples.
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For the next terms in the expansion, you can choose mostly x's with a single dx since there are
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n radicals from which you could have chosen that single dx. This gives us n separate terms all of which include
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n-1 x's times a dx giving a value of x(n-1) times dx. This is analogous to how the majority of
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the new area in the square came from those two bars each with area x * dx,
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or how the bulk of the new volume in the cube came from those three thin squares.
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Each of which had a volume of x2 times dx. There will be many other terms of this expansion
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but all of them are just going to be some multiple of dx2 so we can safely ignore them.
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And what that means is that all but a negligible portion of the increase in the output
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comes from n copies of this x to the (n-1) * dx - that's what it means.
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For the derivative of xn to be n * x(n-1) and even though, like I said in practice, you'll find
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yourself performing this derivative quickly and symbolically imagining the exponent,
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hopping down to the front. Every now and then it's nice to just step back and remember why these rules work.
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Not just because it's pretty, and not just because it helps remind us that math actually makes sense
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and isn't just a pile of formulas to memorize. But because it flexes that very important muscle of
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thinking about derivatives in terms of tiny nudges. As another example think of the function f(x) = 1/x.
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Now, on the one hand you could just blindly try applying the power rule since 1/x
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is the same as writing x(-1). That would involve letting the negative 1 hop down in front
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leaving behind 1 less than itself which is -2. But let's have some fun and see if we can reason about this
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geometrically rather than just plugging it through some formula. The value 1/x is asking what number
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multiplied by x equals 1, so here's how I'd like to visualize it - imagine a little
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rectangular puddle of water sitting in two dimensions whose area is 1, and let's say that its width is x.
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Which means that the height has to be 1/x since the total area of it is 1, so if x was
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stretched out to 2 then that height is forced down to 1/2 and if you increased x up to 3
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then the other side has to be squished down to 1/3 this is a nice way to think about the graph of 1/x.
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By the way, if you think of this with x of the puddle as being in the xy plane then that corresponding output 1/x,
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the height of the graph above that point is whatever the height of your puddle has to be to maintain an area of 1.
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So with this visual in mind for the derivative, imagine nudging up that value of x by some tiny amount,
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some tiny dx. How must the height of this rectangle change so that the area of the puddle remains
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constant at 1 - That is increasing the width by dx, add some new area to the right here so the puddle
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has to decrease in height by some d(1/x) so that the area lost off of that top cancels out the area gained.
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You should think of that d(1/x) as being a negative amount. By the way since it's decreasing the height
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of the rectangle and you know what I'm
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going to leave the last few steps here
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for you for you to pause and ponder and
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work out an ultimate expression and once
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you reason out what D of 1 over X / DX
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should be I want you to compare it to
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what you would have gotten if you had
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just blindly applied the power rule
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purely symbolically
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to
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X to the negative one and while I'm
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encouraging a deposit ponder here's
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another fun challenge if you're feeling
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up to it see if you can reason through
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what the derivative of the square root
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of x should be
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to finish things off I want to tackle
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one more type of function trigonometric
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functions and in particular let's focus
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on the sine function so for this section
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I'm going to assume that you're already
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familiar with how to think about trig
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functions using the unit circle the
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circle with the radius one centered at
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the origin
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for a given value of theta like say zero
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point eight you imagine yourself walking
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around the circle starting from the
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rightmost point until you've traversed
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that distance of zero point eight in arc
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length this is the same thing as saying
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that the angle right here is exactly
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theta radians
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since the circle has a radius of one
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then what sine of theta means is the
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height of that point above the x-axis
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and as your theta value increases and
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you walk around the circle your height
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Bob's up and down between negative one
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and one so when you graph sine of theta
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versus theta you get this wave pattern
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the quintessential wave pattern
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and just from looking at this graph we
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can start to get a feel for the shape of
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the derivative of the sine the slope at
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zero is something positive since sine of
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theta is increasing there and as we move
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to the right and sine of theta
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approaches its peak that slope goes down
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to zero then the slope is negative for a
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little while while the sine is
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decreasing before coming back up to zero
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as the sine graph levels out and as you
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continue thinking this through and
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drawing it out if you're familiar with
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the graph of trig functions you might
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guess that this derivative graph should
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be exactly cosine of theta since all the
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peaks and valleys line up perfectly with
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where the peaks and valleys for the
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cosine function should be and spoiler
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alert the derivative is in fact the
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cosine of theta but aren't you a little
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curious about why it's precisely cosine
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of theta I mean you could have all sorts
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of functions with peaks and valleys at
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the same points that have roughly the
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same shape but who knows maybe the
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derivative of sine could have turned out
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to be some entirely new type of function
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that just happens to have a similar
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shape well just like the previous
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examples a more exact understanding of
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the derivative requires looking at what
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the function actually represents rather
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than looking at the graph of the
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function so think back to that walk
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around the unit circle having traversed
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an arc with length theta and thinking
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about sine of theta as the height of
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that point
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now zoom in to that point on the circle
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and consider a slight nudge of D theta
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along their circumference a tiny step in
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your walk around the unit circle how
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much does that tiny step change the sign
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of theta how much does this increase D
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theta of arc length increase the height
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above the x-axis
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well zoomed in close enough the circle
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basically looks like a straight line in
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this neighborhood so let's go ahead and
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think of this right triangle where the
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hypotenuse of that right triangle
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represents the nudge D theta along the
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circumference and that left side here
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represents the change in height the
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resulting D sine of theta now this tiny
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triangle is actually similar to this
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larger triangle here with the defining
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angle theta and whose hypotenuse is the
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radius of the circle with length one
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specifically this little angle right
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here is precisely equal to theta radians
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now think about what the derivative of
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sine is supposed to mean it's the ratio
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between that D sine of theta the tiny
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change to the height divided by D theta
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the tiny change to the input of the
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function and from the picture we can see
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that that's the ratio between the length
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of the side adjacent to the angle theta
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divided by the hypotenuse well let's see
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adjacent divided by hypotenuse that's
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exactly what the cosine of theta means
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that's the definition of the cosine
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so this gives us two different really
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nice ways of thinking about how the
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derivative of sine is cosine one of them
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is looking at the graph and getting a
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loose feel for the shape of things based
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on thinking about the slope of the sine
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graph at every single point and the
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other is a more precise line of
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reasoning looking at the unit circle
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itself for those of you that like to
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pause and ponder see if you can try a
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similar line of reasoning to find what
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the derivative of the cosine of theta
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should be in the next video I'll talk
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about how you can take derivatives of
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functions who combine simple functions
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like these ones either as sums or
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products or function compositions things
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like that and similar to this video the
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goal is going to be to understand each
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one geometrically in a way that makes it
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intuitively reasonable and somewhat more
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memorable as you know by now there are
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many people to thank for this series and
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one group I'd like to call out
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specifically is brilliant org I think
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explanations but math is not a spectator
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those intuitions is with your own
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explorations and problem solving
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speaking as someone who's worked on
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creating those kinds of sequences before
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would also recommend looking at their
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sequences
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you
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you