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2.5 The Chain Rule

2.5 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. Example:.

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2.5 The Chain Rule

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  1. 2.5 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

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

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

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

  5. 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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