Can I use the chain rule along with the product or quotient rule?

Yes! In fact, you often have to. If you are finding the derivative of $x^2 \sin(3x)$, you must use the product rule for the main expression, and when it comes time to take the derivative of $\sin(3x)$, you use the chain rule.

Do I always have to use the chain rule?

Technically, you are always using it. Even for a basic function like $y = \sin(x)$, the inside function is $x$. The derivative of $x$ is 1. So using the chain rule gives $\cos(x) \cdot 1$, which is just $\cos(x)$. We usually just skip writing the "times 1" step.

What if there are three nested functions?

You just keep chaining! If you have a function like $y = \sin(\cos(x^2))$, you peel it layer by layer. Differentiate the outermost function, multiply by the derivative of the middle function, and finally multiply by the derivative of the innermost function.

When do you use the chain rule?

You use the chain rule whenever you need to take the derivative of a composite function. A composite function is simply a function that has another function tucked inside it, usually written as $f(g(x))$ . If you can identify an "outside" function and an "inside" function, you need the chain rule.

Think of it like opening a set of Russian nesting dolls. To get to the inner doll, you first have to open the outer doll. In calculus, to differentiate the whole expression, you differentiate the outside function first (leaving the inside alone), and then you multiply that result by the derivative of the inside function.

What is the chain rule?

The chain rule is a formula for computing the derivative of the composition of two or more functions. Mathematically, if you have a function $y = f(g(x))$ , the chain rule states that the derivative is $y' = f'(g(x)) \cdot g'(x)$ . You take the derivative of the outside function $f$ , keep the inside function $g(x)$ exactly as it is, and then multiply by the derivative of the inside function $g'(x)$ .

How to recognize when to use it

You can spot a chain rule problem by looking for grouped expressions. Look for parentheses raised to a power, such as $(2x + 1)^3$ . Look for trigonometric functions with expressions other than just $x$ inside them, like $\sin(5x^2)$ . You should also look out for expressions under a square root, or natural logs of polynomials, like $\ln(x^2 + 4)$ . If you can say "a function of a function," the chain rule is your tool.

Where students slip up

The most common mistake students make is differentiating both the inside and the outside functions at the same time. For example, when differentiating $\sin(x^3)$ , a student might write $\cos(3x^2)$ . This is incorrect because it changes the "inside" during the first step. The correct approach leaves the inside alone first: $\cos(x^3)$ , and then multiplies by the derivative of the inside: $\cos(x^3) \cdot 3x^2$ .

Worked through

Find the derivative of $y = (3x^2 - 5)^4$ .

First, identify the outside function and the inside function. The outside function is "something raised to the power of 4" and the inside function is $3x^2 - 5$ .

Step 1: Differentiate the outside function using the power rule, leaving the inside alone. This gives us $4(3x^2 - 5)^3$ .

Step 2: Find the derivative of the inside function, $3x^2 - 5$ . Using the power rule, the derivative is $6x$ .

Step 3: Multiply them together. $y' = 4(3x^2 - 5)^3 \cdot 6x$ .

Step 4: Simplify by multiplying the terms on the outside. $y' = 24x(3x^2 - 5)^3$ .

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Where this comes from: Stewart, James. Calculus: Early Transcendentals. · OpenStax. Calculus Volume 1, Chapter 3.6: The Chain Rule.