Chain Rule

Concept Problem 1 Problem 2 Problem 3 Problem 4

Use the chain rule to find the derivative of the composite of two functions--the derivative of the "outside" function multiplied by the derivative of the "inside" function. The chain rule is related to the product rule and the quotient rule, which gives the derivative of the quotient of two functions.

I want to talk about the chain rule. The chain rule is how we differentiate composite functions.

Now let me remind you what composite functions are. I have three examples here. Composite functions are basically when you take one function and stuff it inside another. Like here, I have the function X squared plus 3X plus 4 and then I'm taking that whole function and raising it to the 12th power. Here, I have 4X squared plus 9 and that's being put inside the square root function. And this last function has 2X cubed minus 5 inside the function E to the X.

It's really important to recognize with composite functions that there is an inside function and an outside function. So to highlight those differences between inside and outside, I've made the inside functions always blue here. And the outside functions are always red. Each of these functions could be expressed as F of G of X where G of X is sort of a general name I'm giving to the inside function. F of of X is the outside. How do you differentiate these? The answer is the chain rule.

This is the chain rule. So the derivative of F of G of X is going to be F prime of G of X times G prime of X. That's the rule we're going to use. You differentiate the outside function first, leave the inside function alone, then multiply by the derivative of the inside function.

Let's try that out on this problem. H of X is -- it's the first of our examples over there. So H prime is going to be -- and according to this formula I differentiate the outside function first. Here the outside function is something to the 12th. So the derivative is going to be 12 times something to the 12 minus 1 or 11. Now, the inside remains unchanged. The G of X is unchanged in this first part. So I just write X squared plus 3X plus 4.

But we're not done. We still have to multiply by the derivative of the inside part. This is the inside part. Its derivative is 2X plus 3. Now we're done.

So this is the chain rule. You differentiate the outside function first, leave the inside alone, and then multiply by the derivative of the inside function.