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3.1 The Chain Rule

Part of Differentiation: Composite, Implicit, and Inverse Functions in AP® Calculus AB. Read the opening section, then create a free account for the full interactive lesson, practice, and progress tracking.

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The chain rule is used to differentiate a composite function, which is a function inside another function. If
y=f(g(x))y=f(g(x))
,
then the derivative is
y′=f′(g(x))⋅g′(x)y'=f'(g(x)) \cdot g'(x)
.
In words, you differentiate the outside function while keeping the inside unchanged, and then multiply by the derivative of the inside. This rule is essential because many important functions in calculus are composites, such as
(3x2+1)5,sin⁡(2x),ex3(3x^2+1)^5, \sin(2x), e^{x^3}
.

A good way to recognize when the chain rule is needed is to ask whether one function has been plugged into another. If the answer is yes, then the expression is likely composite. You should get used to identifying the outer function and the inner function before differentiating. The most common mistake is to differentiate only the outside and forget to multiply by the derivative of the inside. Since the chain rule appears constantly in later calculus, mastering this lesson is one of the biggest turning points in the course.

In this unit

6 lessons

  1. 3.1The Chain RuleThis lesson
  2. 3.2Implicit DifferentiationPreview lesson
  3. 3.3Differentiating Inverse FunctionsPreview lesson
  4. 3.4Differentiating Inverse Trigonometric FunctionsPreview lesson
  5. 3.5Selecting Procedures for Calculating DerivativesPreview lesson
  6. 3.6Calculating Higher-Order DerivativesPreview lesson

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