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In this section, we will investigate how to take the derivative of a function that is the composition of multiple functions.
Section 3.2 The Chain Rule
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The Idea
We know:
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The Idea
We know:
But what about:
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The Idea
These are all composition functions – that is multiple functions chained together. To take the derivative of such functions, we need to extend the “basic” rules with the chain rule.
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TheoremThe Chain Rule
Let
Then:
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TheoremThe Chain Rule
Let
Then:
What this really says:
We take the derivative of the “outside” function (leaving the “inside” function alone), and then multiply the result by the derivative of the “inside” function.
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Example 1
Find the derivative of the function
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Example 2
Find the derivative of the function
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Example 3
Find the derivative of the function
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Example 4
Find the derivative of the function
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Example 5
Find the derivative of the function
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Example 6
Find the derivative of the function
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Example 7Numerical Example
Suppose we know:
Calculate