Lect20 - Derivatives of Inverse Functions If y = f -1 ( x)...

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92.131 Lecture 20 1 of 10 Ronald Brent © 2008 All rights reserved. Derivatives of Inverse Functions If ) ( 1 x f y = , then ) ( y f x = . Using implicit differentiation: dx dy y f y f x d d x x d d ) ( )) ( ( ) ( = = so that dx dy y f ) ( 1 = and )) ( ( 1 ) ( 1 1 x f f y f dx dy = = . Example: a) Let 2 3 ) ( 1 + = = x x f y , then 3 2 ) ( = y y f . 3 1 ) ( = x f and 3 3 1 1 = = dx dy
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92.131 Lecture 20 2 of 10 Ronald Brent © 2008 All rights reserved. b) Let 4 ) ( 2 1 + = = x x f y then 4 ) ( = y y f . Since 4 2 1 ) ( = y y f we have x x x x dx dy 2 2 4 ) 4 ( 2 4 ) 4 ( 2 1 1 2 2 2 = = + = + = c ) L e t 2 ) ( 1 x e x f y = = then y y f ln ) ( = . S i n c e y y y f ln 2 1 ) ( = we have 2 2 2 2 ln 2 ln 2 1 1 2 x x e x x e y y y y dx dy = = = =
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92.131 Lecture 20 3 of 10 Ronald Brent © 2008 All rights reserved. Derivatives of Logarithmic Functions Recall that: x y ln = , if and only if y e x = . Using implicit differentiation: = = ) ( ) ( y e x d d x x d d gives y e y = 1 so x e y y 1 1 = = Theorem 1) x x 1 ) (ln = 2) x a x a ) (ln 1 ) (log =
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92.131 Lecture 20
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Lect20 - Derivatives of Inverse Functions If y = f -1 ( x)...

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