The Chain Rule (2.4) September 30th, 2016.

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Presentation transcript:

The Chain Rule (2.4) September 30th, 2016

I. the chain rule Thm. 2.10: The Chain Rule: If y = f(u) is a differentiable function of u and u = g(x) is a differentiable function of x, then y=f(g(x)) is a differentiable function of x and -or-

Proof of Thm. 2.10: The Chain Rule Prove

Alternate Proof of the Quotient Rule Use the chain rule and the product rule to prove

II. The general power rule Thm. 2.11: The General Power Rule: If , where u is a differentiable function of x and n is a rational number, then -or-

Ex. 1: Differentiate each of the following functions. b) c) d) e)

Ex. 2: Use the graphs of f(x) and g(x) below to evaluate each of the following given that and .