# lec5 - MIT OpenCourseWare http:/ocw.mit.edu 18.01 Single...

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MIT OpenCourseWare http://ocw.mit.edu 18.01 Single Variable Calculus Fall 2006 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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± ± ± Lecture 5 18.01 Fall 2006 Lecture 5 Implicit Diﬀerentiation and Inverses Implicit Diﬀerentiation d Example 1. ( x a ) = ax a 1 . dx We proved this by an explicit computation for a = 0 , 1 , 2 ,... . From this, we also got the formula for a = 1 , 2 ,... . Let us try to extend this formula to cover rational numbers, as well: m m a = ; y = x n where m and n are integers. n We want to compute dy . We can say y n = x m so ny n 1 dy = mx m 1 . Solve for dy : dx dx dx dy = m x m 1 dx n y n 1 ( m We know that y = x n ) is a function of x . dy = m x m 1 dx n y n 1 m x m 1 = n ( x m/n ) n 1 m x m 1 = n x m ( n 1) /n = x ( m 1) m ( n n 1) m n m n ( m 1) m ( n 1) = x n n m nm n nm + m = x n n m m n = x n n n dy m m So, =
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## This note was uploaded on 01/18/2012 for the course MATH 18.01 taught by Professor Brubaker during the Fall '08 term at MIT.

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lec5 - MIT OpenCourseWare http:/ocw.mit.edu 18.01 Single...

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