Lec5 - 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

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Unformatted text preview: 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, = x n- 1 dx n This is the same answer as we were hoping to get! Example 2. Equation of a circle with a radius of 1: x 2 + y 2 = 1 which we can write as y 2 = 1- x 2 ....
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This note was uploaded on 02/27/2009 for the course MATH 155b taught by Professor Staff during the Fall '08 term at Vanderbilt.

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Lec5 - 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

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