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must must - Math 140(07 Homework Set#4 Ae Ja Yee 3.6...

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Math 140 (07), Homework Set #4 Ae Ja Yee 3.6 Implicit Differentiation Find dy dx by implicit differentiation. 14. y sin( x 2 ) = x sin( y 2 ) Solution. y 0 sin( x 2 ) + 2 xy cos( x 2 ) = sin( y 2 ) + 2 xyy 0 cos( y 2 ) y 0 (sin( x 2 ) - 2 xy cos( y 2 )) = sin( y 2 ) - 2 xy cos( x 2 ) y 0 = sin( y 2 ) - 2 xy cos( x 2 ) sin( x 2 ) - 2 xy cos( y 2 ) 16. x + y = 1 + x 2 y 2 Solution. ( x + y ) 1 / 2 = 1 + x 2 y 2 1 2 ( x + y ) - 1 / 2 (1 + y 0 ) = 2 xy 2 + 2 x 2 yy 0 y 0 ( 1 2 ( x + y ) - 1 / 2 - 2 x 2 y ) = 2 xy 2 - 1 2 ( x + y ) - 1 / 2 y 0 = 2 xy 2 - 1 2 ( x + y ) - 1 / 2 1 2 ( x + y ) - 1 / 2 - 2 x 2 y = 4 xy 2 x + y - 1 1 - 4 x 2 y x + y 22. If g ( x ) + x sin g ( x ) = x 2 , find g 0 (0). Solution. g 0 ( x ) + sin g ( x ) + xg 0 ( x ) cos g ( x ) = 2 x g 0 ( x )(1 + x cos g ( x )) = 2 x - sin g ( x ) g 0 ( x ) = 2 x - sin g ( x ) 1 + x cos g ( x ) g 0 (0) = - sin g (0) Note that g (0) = 0, so g 0 (0) = 0. 30. Use implicit differentiation to find an equation of the tangent line to the curve at the given point. y 2 ( y 2 - 4) = x 2 ( x 2 - 5) , (0 , - 2) 1
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Solution. y 4 - 4 y 2 = x 4 - 5 x 2 4 y 3 y 0 - 8 yy 0 = 4 x 3 - 10 x y 0 (4 y 3 - 8 y ) = 4 x 3 - 10 x y 0 = 2 x 3 - 5 x 2 y 3 - 4 y (0 , - 2) = 0 Thus, the tangent line is y = - 2 36. Find y 00 by implicit differentiation.
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