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Objectives

  • State the chain rule for the composition of two functions.
  • Apply the chain rule together with the power rule.
  • Apply the chain rule and the product/quotient rules correctly in combination when both are necessary.
  • Recognize the chain rule for a composition of three or more functions.
  • Describe the proof of the chain rule.

Summary

A composite function is one where the input variable x first passes through one function, and then the resulting output passes through another. For example, the function h ( x ) = 2 sin ⁡ ( x ) is composite since x ⟶ sin ⁡ ( x ) ⟶ 2 sin ⁡ ( x ) .

Given a composite function C ( x ) = f ( g ( x ) ) where f and g are differentiable functions, the chain rule tells us that

C ′ ( x ) = f ′ ( g ( x ) ) g ′ ( x ) .

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