Remember, in our three-lines setup, when we took the derivative of sine on the bottom, we expanded the size of the nudge d ( sin ) d(\sin) d ( sin ) as cos ( h ) ⋅ d h \cos(h) \cdot dh cos ( h ) ⋅ d h. But it’s also an important reflection of what this derivative of the outer function actually represents. On the symbolic level, this serves as a reminder that you still plug in the inner function to this derivative. Notice, for the derivative of g g g, I’m writing it as d d h \frac d x d . The chain rule for the function f ( x ) = g ( h ( x ) ) f(x) = g(h(x)) f ( x ) = g ( h ( x )) I’ll leave it to you to pause and ponder to verify that this makes sense. The derivative is just that same constant times the derivative of the function, in this case 2 ⋅ cos ( x ) 2 \cdot \cos(x) 2 ⋅ cos ( x ). Constant multiplicationīy the way, I should mention that if you multiply by a constant, say 2 ⋅ sin ( x ) 2 \cdot \sin(x) 2 ⋅ sin ( x ), things end up much simpler. "Left d right" is the area of this bottom rectangle, and “right d left” is the area of this rectangle on the right. Out of context, this feels like kind of a strange rule, but when you think of this adjustable box you can actually see how those terms represent slivers of area. Then you add "right d left": the right function, x 2 x^2 x 2, times the derivative of the left, cos ( x ) \cos(x) cos ( x ). In this example, sin ( x ) ⋅ x 2 \sin(x) \cdot x^2 sin ( x ) ⋅ x 2, "left d right" means you take the left function, in this case g ( x ) = sin ( x ) g(x) = \sin(x) g ( x ) = sin ( x ), times the derivative of the right, h ( x ) = x 2 h(x) = x^2 h ( x ) = x 2, which gives 2 x 2x 2 x. Generic Product Rule where f ( x ) = g ( x ) h ( x ) f(x) = g(x)h(x) f ( x ) = g ( x ) h ( x ).Ī common mnemonic for the product rule is to say in your head "left d right, right d left". So, the question is, if you know the derivatives of two functions, what is the derivative of their sum, of their product, and of the function compositions between them? Sum rule But as long as you know how derivatives play with those three types of combinations, you can always just take it step by step and peel through the layers. Most functions you come across just involve layering on these three types of combinations, with no bound on how monstrous things can become. Likewise, dividing functions is really just the same as plugging one into the function 1 / x 1/x 1/ x, then multiplying. Sure, you could say subtracting them, but that’s really just multiplying the second by − 1 -1 − 1, then adding. This really boils down into three basic ways to combine functions together: Adding them, multiplying them, and putting one inside the other also known as composing them. Most functions you use to model the world involve mixing, combining and tweaking these simple functions in some way so our goal now is to understand how to take derivatives of more complicated combinations where again, I want you to have a clear picture in mind for each rule. In the last videos I talked about the derivatives of simple functions, things like powers of x x x, sin ( x ) \sin(x) sin ( x ), and exponentials, the goal being to have a clear picture or intuition to hold in your mind that explains where these formulas come from. “Using the chain rule is like peeling an onion: you have to deal with each layer at a time, and if it is too big you will start crying.”
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