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Mastering the Chain Rule: Differentiation Techniques Simplified

Learn how to apply the Chain Rule efficiently in calculus to find derivatives of composite functions. Practice with examples and grasp the concept of differentiating the outer and inner functions easily.

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Mastering the Chain Rule: Differentiation Techniques Simplified

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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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