12.3 The Chain and General Power Rules
7) The Chain Rule
If ℎ() = [()], then: ℎ′ () = �()� = ′(()) ∙ ′() Equivalently, if we let = () we would have ℎ() = () and thus = ℎ() = (), then: = ∙ Many composite functions have the form ℎ() = [()] where n is a real number. In these cases a shortcut
formula called the general power rule can be used to find ℎ′().
The General Power Rule If the function f is differentiable and ℎ() = [()] (n is a real number), then: ℎ′ () = [()] = [()]−1 ∙ ′() Example 1 Find ′() for () = ( 2 − 4 + 3)5 using (a) the definition of the chain rule and (b) the general power rule. Example 2
Find the derivatives of the following functions:
3 a) () = √2 6 − 5 2 b) () = 2 2 (7 3 − 1)8 − 5 13 c) ℎ() = �2 + 1� d) () = (6 − 1)4 (2 2 + 8)5 e) () = f) ℎ() = � 5 − � 9 (3 + 2)7 (7 2 4 − 3)10

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