12.3 The Chain and General Power Rules.pdf

12.3 The Chain and General Power Rules.pdf - 12.3 The Chain...

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12.3 The Chain and General Power Rules 7) The Chain Rule If [ ] ) ( ) ( x f g x h = , then: = = h x d dx g f x g f x f x ( ) ( ( )) ( ( )) ( ) Equivalently, if we let ) ( x f U = we would have ) ( ) ( U g x h = and thus ) ( ) ( U g x h y = = , then: dy dx dy dU dU dx = Many composite functions have the form [ ] n x f x h ) ( ) ( = where n is a real number. In these cases a shortcut formula called the general power rule can be used to find h x ( ) . The General Power Rule If the function f is differentiable and [ ] n x f x h ) ( ) ( = ( n is a real number), then: = = h x d dx f x n f x f x n n ( ) [ ( )] [ ( )] ( ) 1 Example 1 Find
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