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Unformatted text preview: y = [ g ( x )] n , we get dy dx = dy du du dx = nu n1 du dx = n [ g ( x )] n1 g ( x ) . Here, we wrote y = u n where u = g ( x ). Example 2. Diﬀerentiate f ( x ) = 3 √ x 2 + x + 1. Solution f ( x ) = 1 3 ( x 2 + x + 1)2 3 · (2 x + 1) We can use the chain rule to diﬀerentiate an exponential function with any base. If we use a x = ( e ln a ) x = e (ln a ) x , then d dx ( a x ) = d dx e (ln a ) x = e (ln a ) x d dx (ln a ) x = e (ln a ) x · ln a = a x ln a 1 2 CHAIN RULE If we have a longer chain, we can use the chain rule twice or more. Example 3. Diﬀerentiate f ( x ) = sin(sin(sin x )). Solution f ( x ) = cos(sin(sin x )) d dx sin(sin x ) = cos(sin(sin x )) cos(sin x ) d dx (sin x ) = cos(sin(sin x )) cos(sin x ) cos x...
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This note was uploaded on 03/27/2012 for the course MATH 1a taught by Professor Benedictgross during the Fall '09 term at Harvard.
 Fall '09
 BenedictGross
 Chain Rule

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