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15:56
Visualizing the chain rule and product rule | Chapter 4, Essence of calculus
3Blue1Brown
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May 12, 2026
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Transcript
0:14
In the last videos I talked about the derivatives of simple functions,
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and the goal was to have a clear picture or intuition to hold in
0:22
your mind that actually explains where these formulas come from.
0:26
But most of the functions you deal with in modeling the world involve mixing,
0:31
combining, or tweaking these simple functions in some other way,
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so our next step is to understand how you take derivatives of more complicated
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combinations.
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Again, I don't want these to be something to memorize,
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0:43
I want you to have a clear picture in mind for where each one comes from.
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Now, this really boils down into three basic ways to combine functions.
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You can add them together, you can multiply them,
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and you can throw one inside the other, known as composing them.
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Sure, you could say subtracting them, but really that's just
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multiplying the second by negative one and adding them together.
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Likewise, dividing functions doesn't really add anything,
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because that's the same as plugging one inside the function, one over x,
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and then multiplying the two together.
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So really, most functions you come across just involve layering
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together these three different types of combinations,
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though there's not really a bound on how monstrous things can become.
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But as long as you know how derivatives play with just those three combination types,
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you'll always be able to take it step by step and peel through
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the layers for any kind of monstrous expression.
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So the question is, if you know the derivative of two functions,
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what is the derivative of their sum, of their product,
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and of the function composition between them?
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The sum rule is easiest, if somewhat tongue-twisting to say out loud.
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The derivative of a sum of two functions is the sum of their derivatives.
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But it's worth warming up with this example by really thinking through
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what it means to take a derivative of a sum of two functions,
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since the derivative patterns for products and function composition won't
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be so straightforward, and they're going to require this kind of deeper thinking.
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For example, let's think about this function f of x equals sine of x plus x squared.
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It's a function where, for every input, you add together
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the values of sine of x and x squared at that point.
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For example, let's say at x equals 0.5, the height of the sine
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graph is given by this vertical bar, and the height of the x
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squared parabola is given by this slightly smaller vertical bar.
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And their sum is the length you get by just stacking them together.
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For the derivative, you want to ask what happens as you nudge that input slightly,
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maybe increasing it up to 0.5 plus dx.
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The difference in the value of f between those two places is what we call df.
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And when you picture it like this, I think you'll agree that the total
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change in the height is whatever the change to the sine graph is,
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what we might call d sine of x, plus whatever the change to x squared is, dx squared.
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We know that the derivative of sine is cosine, and remember what that means.
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It means that this little change, d sine of x, is about cosine of x times dx.
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It's proportional to the size of our initial nudge dx,
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and the proportionality constant equals cosine of whatever input we started at.
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Likewise, because the derivative of x squared is 2x,
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the change in the height of the x squared graph is 2x times whatever dx was.
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So rearranging df divided by dx, the ratio of the tiny change to
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the sum function to the tiny change in x that caused it,
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is indeed cosine of x plus 2x, the sum of the derivatives of its parts.
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But like I said, things are a bit different for products,
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and let's think through why in terms of tiny nudges again.
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In this case, I don't think graphs are our best bet for visualizing things.
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Pretty commonly in math, at a lot of levels of math really,
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if you're dealing with a product of two things,
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it helps to understand it as some kind of area.
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In this case, maybe you try to configure some mental setup
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of a box where the side lengths are sine of x and x squared.
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But what would that mean?
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Well, since these are functions, you might think of those sides as adjustable,
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dependent on the value of x, which maybe you think of as this
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number that you can just freely adjust up and down.
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So getting a feel for what this means, focus on
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that top side who changes as the function sine of x.
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As you change this value of x up from 0, it increases up to
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a length of 1 as sine of x moves up towards its peak,
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and after that it starts to decrease as sine of x comes down from 1.
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And in the same way, that height there is always changing as x squared.
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So f of x, defined as the product of these two functions, is the area of this box.
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And for the derivative, let's think about how
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a tiny change to x by dx influences that area.
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What is that resulting change in area df?
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Well, the nudge dx caused that width to change by some small d sine of x,
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and it caused that height to change by some dx squared.
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And this gives us three little snippets of new area,
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a thin rectangle on the bottom whose area is its width, sine of x,
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times its thin height, dx squared.
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And there's this thin rectangle on the right, whose area is its height,
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x squared, times its thin width, d sine of x.
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And there's also this little bit in the corner, but we can ignore that.
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Its area is ultimately proportional to dx squared,
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and as we've seen before, that becomes negligible as dx goes to zero.
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I mean, this whole setup is very similar to what I showed last video,
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with the x squared diagram.
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And just like then, keep in mind that I'm using somewhat beefy
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changes here to draw things, just so we can actually see them.
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But in principle, dx is something very very small,
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and that means that dx squared and d sine of x are also very very small.
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So, applying what we know about the derivative of sine and of x squared,
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that tiny change, dx squared, is going to be about 2x times dx.
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And that tiny change, d sine of x, well that's going to be about cosine of x times dx.
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As usual, we divide out by that dx to see that the ratio we want, df divided by dx,
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is sine of x times the derivative of x squared,
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plus x squared times the derivative of sine.
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And nothing we've done here is specific to sine or to x squared.
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This same line of reasoning would work for any two functions, g and h.
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And sometimes people like to remember this pattern with
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a certain mnemonic that you kind of sing in your head.
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Left d right, right d left.
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In this example, where we have sine of x times x squared, left d right,
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means you take that left function, sine of x, times the derivative of the right,
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in this case 2x.
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Then you add on right d left, that right function,
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x squared, times the derivative of the left one, cosine of x.
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Now out of context, presented as a rule to remember,
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I think this would feel pretty strange, don't you?
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But when you actually think of this adjustable box,
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you can see what each of those terms represents.
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Left d right is the area of that little bottom rectangle,
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and right d left is the area of that rectangle on the side.
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By the way, I should mention that if you multiply by a constant,
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say 2 times sine of x, things end up a lot simpler.
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The derivative is just the same as the constant multiplied by
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the derivative of the function, in this case 2 times cosine of x.
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I'll leave it to you to pause and ponder and verify that makes sense.
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Aside from addition and multiplication, the other common way to combine functions,
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and believe me, this one comes up all the time,
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is to shove one inside the other, function composition.
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For example, maybe we take the function x squared and shove it
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inside sine of x to get this new function, sine of x squared.
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What do you think the derivative of that new function is?
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To think this one through, I'll choose yet another way to visualize things,
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just to emphasize that in creative math, we've got lots of options.
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I'll put up three different number lines, the top one is going to hold the value of x,
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the second one is going to hold the x squared,
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and the third line is going to hold the value of sine of x squared.
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That is, the function x squared gets you from line 1 to line 2,
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and the function sine gets you from line 2 to line 3.
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As I shift around this value of x, maybe moving it up to the value 3,
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that second value stays pegged to whatever x squared is, in this case moving up to 9.
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That bottom value, being sine of x squared, is
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going to go to whatever sine of 9 happens to be.
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So, for the derivative, let's again start by nudging that x value by some little dx.
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I always think that it's helpful to think of x as starting
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at some actual concrete number, maybe 1.5 in this case.
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The resulting nudge to that second value, the change in x squared caused by such a dx,
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is dx squared.
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We could expand this like we have before, as 2x times dx,
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which for our specific input would be 2 times 1.5 times dx,
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but it helps to keep things written as dx squared, at least for now.
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In fact, I'm going to go one step further, give a new name to this x squared,
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maybe h, so instead of writing dx squared for this nudge, we write dh.
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This makes it easier to think about that third value, which is now pegged at sine of h.
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Its change is d sine of h, the tiny change caused by the nudge dh.
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By the way, the fact that it's moving to the left while the dh bump is going to the right
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just means that this change, d sine of h, is going to be some kind of negative number.
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Once again, we can use our knowledge of the derivative of the sine.
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This d sine of h is going to be about cosine of h times dh.
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That's what it means for the derivative of sine to be cosine.
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Unfolding things, we can replace that h with x squared again,
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so we know that the bottom nudge will be a size of cosine of x squared times dx squared.
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Let's unfold things even further.
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That intermediate nudge dx squared is going to be about 2x times dx.
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It's always a good habit to remind yourself of
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what an expression like this actually means.
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In this case, where we started at x equals 1.5 up top,
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this whole expression is telling us that the size of the nudge on that third
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line is going to be about cosine of 1.5 squared times 2 times 1.5 times whatever
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the size of dx was.
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It's proportional to the size of dx, and this
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derivative is giving us that proportionality constant.
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Notice what we came out with here.
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We have the derivative of the outside function,
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and it's still taking in the unaltered inside function,
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and then multiplying it by the derivative of that inside function.
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Again, there's nothing special about sine of x or x squared.
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If you have any two functions, g of x and h of x,
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the derivative of their composition, g of h of x,
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is going to be the derivative of g evaluated on h, multiplied by the derivative of h.
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This pattern right here is what we usually call the chain rule.
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Notice for the derivative of g, I'm writing it as dg dh instead of dg dx.
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On the symbolic level, this is a reminder that the thing you plug
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into that derivative is still going to be that intermediary function h.
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But more than that, it's an important reflection of what
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this derivative of the outer function actually represents.
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Remember, in our three line setup, when we took the derivative of the sine on
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that bottom, we expanded the size of that nudge, d sine, as cosine of h times dh.
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This was because we didn't immediately know how
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the size of that bottom nudge depended on x.
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That's kind of the whole thing we were trying to figure out.
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But we could take the derivative with respect to that intermediate variable, h.
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That is, figure out how to express the size of that nudge on the third
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line as some multiple of dh, the size of the nudge on the second line.
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It was only after that that we unfolded further by figuring out what dh was.
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In this chain rule expression, we're saying, look at the ratio between a tiny change in
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g, the final output, to a tiny change in h that caused it,
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h being the value we plug into g.
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Then multiply that by the tiny change in h, divided
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by the tiny change in x that caused it.
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So notice, those dh's cancel out, and they give us a ratio
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between the change in that final output and the change to the input that,
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through a certain chain of events, brought it about.
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And that cancellation of dh is not just a notational trick.
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That is a genuine reflection of what's going on with the
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tiny nudges that underpin everything we do with derivatives.
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So those are the three basic tools to have in your belt to handle
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derivatives of functions that combine a lot of smaller things.
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You've got the sum rule, the product rule, and the chain rule.
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And I'll be honest with you, there is a big difference between knowing
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what the chain rule is and what the product rule is,
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and actually being fluent with applying them in even the most hairy of situations.
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Watching videos, any videos, about the mechanics of calculus is
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never going to substitute for practicing those mechanics yourself,
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and building up the muscles to do these computations yourself.
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I really wish I could offer to do that for you,
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but I'm afraid the ball is in your court, my friend, to seek out the practice.
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What I can offer, and what I hope I have offered,
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is to show you where these rules actually come from.
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To show that they're not just something to be memorized and hammered away,
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but they're natural patterns, things that you too could have discovered
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just by patiently thinking through what a derivative actually means.
— end of transcript —
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