lec11 - 10. The Chain Rule P. K. Lamm Lecture Notes:...

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10. The Chain Rule P. K. Lamm 10/02/11 (14:44) p. 1 / 5 Lecture Notes: 10. The Chain Rule These classnotes are intended to be supplementary to the textbook and are necessarily limited by the time allotted for classes. For full and precise statements of definitions and theorems, as well as material covering other topics and examples, please consult the textbook. 1. The Chain Rule We have seen some rules for derivatives such as ( x n ) 0 = nx n - 1 , for integer n 6 = 0. The situation changes however when, instead of raising x to a power, we raise a function of x to a power. For example, for g differentiable we have from the product rule that ( g 2 ( x )) 0 = ( g ( x ) · g ( x )) 0 = g ( x ) g 0 ( x ) + g 0 ( x ) g ( x ) . Thus ( g 2 ( x )) 0 = 2 g ( x ) · g 0 ( x ) instead of 2 g ( x ) as we might have expected. Similarly, we may use the formula for the derivative of a product of three factors to obtain ( g 3 ( x )) 0 = ( g ( x ) g ( x ) g ( x )) 0 = g 0 ( x ) g ( x ) g ( x ) + g ( x ) g 0 ( x ) g ( x ) + g ( x ) g ( x ) g 0 ( x ) or that ( g 3 ( x )) 0 = 3 g 2 ( x ) g 0 ( x ) . In fact it’s true that for integer n 6 = 0, ( g n ( x )) 0 = ng n - 1 ( x ) g 0 ( x ) . (1) Let’s introduce some notation. Let f ( u ) = u n , and u ( x ) = g ( x ) . Then with this notation we may write
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This note was uploaded on 04/02/2012 for the course MTH 132 taught by Professor Kihyunhyun during the Fall '10 term at Michigan State University.

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lec11 - 10. The Chain Rule P. K. Lamm Lecture Notes:...

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