The Chain Rule: An Introduction

What the Chain Rule Is For

Many of the derivative rules you learn early in calculus work directly on familiar functions. The power rule handles powers of , while basic derivative rules handle functions such as , , and .

But many functions are more complicated because one function is inside another. These are called composite functions. For example, consider .

The expression has two layers. The inside function is , and the outside function takes that entire expression to the fifth power. The chain rule lets us differentiate one layer at a time.

The Formula

If is a composite function, then:

In words: differentiate the outside function while leaving the inside function alone, then multiply by the derivative of the inside function.

That final multiplication is the key idea. If you differentiate only the outside function and forget the derivative of the inside function, your answer is incomplete.

Thinking in Layers

A useful way to recognize a chain-rule problem is to think about layers. In a function such as , the innermost expression is . The outer operation is raising that entire expression to the fifth power.

You can temporarily call the inside expression . Then the original function becomes . This makes the structure much easier to see.

After differentiating the outside, replace with the original inside function and multiply by its derivative.

A Step-by-Step Method

Identify the outside and inside functions. Ask: "What operation is being performed last?"
Differentiate the outside function, treating the entire inside function as a single object.
Multiply by the derivative of the inside function.
Simplify the result when appropriate.

Worked Examples

Example 1: Power of a Polynomial

Differentiate .

Step 1: Outside function: . Inside function: .

Step 2: Apply the power rule to the outside:

Step 3: Differentiate the inside:

Step 4: Multiply:

Result:

Example 2: Trigonometric Composition

Differentiate .

Step 1: Outside function: . Inside function: .

Step 2: Differentiate the outside:

Step 3: Differentiate the inside:

Step 4: Multiply:

Result:

Example 3: Exponential Composition

Differentiate .

Step 1: Outside function: . Inside function: .

Step 2: The derivative of is itself, so:

Step 3: Differentiate the inside:

Step 4: Multiply:

Result:

Example 4: Square Root as a Power

Differentiate .

Step 1: Rewrite the square root as a fractional power:

Step 2: Differentiate the outside:

Step 3: Differentiate the inside:

Step 4: Multiply and simplify:

Result:

Example 5: Chain Rule Inside the Product Rule

Differentiate . This is closer to the kind of problem that appears on a challenging AP Calculus question because more than one derivative rule is needed.

Step 1: This is a product, so begin with the product rule:

Step 2: The derivative of is:

Step 3: For , use the chain rule. The outside is and the inside is .

Step 4: Therefore:

Result:

How to Recognize a Chain Rule Problem

Look for a function being applied to something more complicated than just . Common examples include:

A particularly useful question is: "What would I have to replace with a single variable to make this look like a basic derivative rule?"

Common Mistakes

Forgetting to multiply by the inside derivative

This is the most common chain rule error. For example, the derivative of is not simply . You must also multiply by the derivative of the inside function.

Differentiating the inside too early

Keep the inside function intact while differentiating the outside. For , first differentiate the outside as though the entire expression were one object. Only afterward multiply by .

Applying the power rule to only part of the expression

In , the exponent applies to the entire quantity . You cannot simply differentiate the and leave the rest unchanged.

Expanding when it is unnecessary

You can sometimes expand a composite expression first, but this is often inefficient and can create unnecessary algebra. The chain rule is usually the cleaner approach when the function has an obvious nested structure.

Practice Problems

Question 1

Differentiate .

Show answer

Question 2

Differentiate .

Show answer

Question 3

Differentiate .

Show answer

Question 4 (Combined Rules)

Differentiate .

Show answer

Test yourself: AP Calculus practice exams

Ready to check what you know? Work an AP-style practice exam, then open the full solutions.

AB Multiple Choice AB Free Response BC Multiple Choice BC Free Response

Want all 1,114 practice problems?

This tutorial covers one topic. Calc Pass covers all of AP Calculus AB & BC: full review, every practice question, and step-by-step solutions to each one.