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Problem Set 07s

# Problem Set 07s - Hung Nguyen Problem Set 07 Solution 3.5...

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Hung Nguyen Problem Set 07 Solution GE 207C 3.5 Implicit Differentiation 1 1. Find dx dy by implicit differentiation a. y x = 8 xy b. tan x y ( ) = 2 x y c. 2 2 3 2 3 = + xy y x x Solution: ( ) ( ) 0 3 2 6 0 3 2 6 0 2 2 2 2 2 3 2 2 3 2 2 3 2 3 3 2 3 = + + = + + = + = + dx dy xy x y xy x dx dy xy y dx dy x xy x xy y x x dx d xy y x x ( ) 2 3 2 3 2 6 y x x y xy x dx dy + = d. ( ) ( ) 2 2 sin sin y x x y = Solution: ( ) ( ) ( ) [ ] ( ) [ ] ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) 2 2 2 2 2 2 2 2 2 2 2 2 cos 2 sin cos 2 sin 2 cos ) sin( sin 2 cos sin sin sin sin x xy y y y x dx dy dx dy y y y dx dy x x x y y x dx d x y dx d y x x y = + = + = = ( ) ( ) ( ) ) cos( 2 sin cos 2 sin 2 2 2 2 y y x x xy y dx dy = e. y x xy 2 1 + = Solution: xy = 1 + x 2 y d dx xy " # \$ % = d dx 1 + x 2 y " # \$ % 1 2 xy ( ) 1/2 x dy dx + y ' ( ) * + , = 0 + x 2 dy dx + y 2 x x 2 xy dy dx + y 2 xy = x 2 dy dx + 2 xy dy dx x 2 x 2 xy ' ( ) ) * + , , = y 2 xy 2 xy = y 4 xy xy 2 xy dy dx 2 x 2 xy x 2 xy ' ( ) ) * + , , = y 4 xy xy 2 xy ( ) ( ) 1 2 4 1 = xy x x xy x y dx dy f. f x ( ) = 4sin 1 x 4 " # \$ % & ' 16 x 2

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Hung Nguyen Problem Set 07 Solution GE 207C 3.5 Implicit Differentiation
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