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Chain Rule in Calculus: Formula & Examples

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Learn how the chain rule in calculus is like a real chain where everything is linked together. Also learn what situations the chain rule can be used in to make your calculus work easier.

What Is the Chain Rule?

The chain rule allows us to differentiate composite functions. What exactly are composite functions? Good question! A composite function is a function whose variable is another function. Look at this example:

A Composite Function
chain rule

The first function is a straightforward function. But the second is a composite function. Do you see how the lone variable x from the first function has been replaced with x^2+1, a function in its own, right? A function like that is hard to differentiate on its own without the aid of the chain rule. But with it, differentiating is a breeze! Let's take a look.

The Formula

Without further ado, here is the formal formula for the chain rule. Don't get scared. It gets simpler once you start using it.

The Chain Rule
chain rule

It may look complicated, but it's really not. All it's saying is that, if you have a composite function and need to take the derivative of it, all you would do is to take the derivative of the function as a whole, leaving the smaller function alone, then you would multiply it with the derivative of the smaller function. Okay. That was a mouthful and thankfully, it's much easier to understand in action, as you will see.

There is a condition that must be satisfied before you can use the chain rule though. It is that both functions must be differentiable at x. Alternately, if you can't differentiate one of the functions, then you can't use the chain rule. It can't help you in those instances.

When to Use It

You will know when you can use it by just looking at a function. If it looks like something you can differentiate, but with the variable replaced with something that looks like a function on its own, then most likely you can use the chain rule.

See if you can see a pattern in these examples.

More Composite Functions
chain rule

I've given you four examples of composite functions. The first and third are examples of functions that are easy to derive. The second and fourth cannot be derived as easily as the other two, but do you notice how similar they look? They look like something you can easily derive, but they have smaller functions in place of our usual lone variable.

How to Use It

Now, let us get into how to actually derive these types of functions. The formula tells us to differentiate the whole thing as if it were a straightforward function that we know how to derive. Then we would multiply it by the derivative of the inside part or the smaller function. Thinking about this, I can make my problems a bit cleaner looking by making a small substitution to change the way I write the function. I can label my smaller inside function with the variable u.

Using Substitution to Simplify the Function
chain rule

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