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2.4 The Chain Rule If f and g are both differentiable and F is the composite function defined by F(x)=f(g(x)), then F is differentiable and F′ is given by the product In Leibniz notation, if y=f(u) and u=g(x) are both differentiable functions, then

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Example: A faster way to write the solution: Differentiate the outer function... …then the inner function

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Another example: It looks like we need to use the chain rule again! derivative of the outer function derivative of the inner function

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Another example: The chain rule can be used more than once. (That’s what makes the “chain” in the “chain rule”!)

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**The Power Rule combined with the Chain Rule**

If n is any real number and u=g(x) is differentiable, then Alternatively, Example:

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The Fundamental Theorem of Calculus is appropriately named because it establishes connection between the two branches of calculus: differential calculus.

The Fundamental Theorem of Calculus is appropriately named because it establishes connection between the two branches of calculus: differential calculus.

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