3.7.2-eg-1 - Example Use implicit differentiation to find y...

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Unformatted text preview: Example Use implicit differentiation to find y 00 ( x ) if ( x,y ( x )) satisfy x 3 + 4 x + y 3 = 17 . In addition, find y 00 ( x ) at the point (2 , 1). Solution: As a reminder that y depends on x , the above equation can be written x 3 + 4 x + ( y ( x )) 3 = 17 . Differentiating both sides of the equation with respect to x , d dx x 3 + 4 x + ( y ( x )) 3 = d dx (17) , leads to 3 x 2 + 4 + 3 y 2 ( x ) y ( x ) = 0 , (1) where we have used the Chain Rule to evaluate d dx ( y ( x )) 3 = 3( y ( x )) 2 y ( x ) . In order to solve equation (1) for y ( x ), first collect all terms involving y ( x ) on the left-hand side of the equation, and all other terms on the right-hand side. That is, 3 y 2 ( x ) y ( x ) =- 3 x 2- 4 , or, A ( x,y ) y = B ( x,y ), (2) where A ( x,y ) = 3 y 2 , and B ( x,y ) =- 3 x 2- 4 . (Note that we have replaced y ( x ) and y ( x ) by their simpler expressions y and y , respectively, in the above.) Solving equation (2) for y results in y = B ( x,y ) A ( x,y ) =...
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3.7.2-eg-1 - Example Use implicit differentiation to find y...

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