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Calculus/Implicit Differentiation

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Expert: - 12/3/2011


Question
Could you please help me solve tan(xy)=xy with implicit differentiation? I got the derivative: sec^2(xy)(y+x(dy/dx))=y+x(dy/dx) and I am not sure what to do now.

Answer
What is left to do is solve for dy/dx.
That is, we have secē(xy)y + secē(xy)(dy/dx) = y + x(dy/dx).

Putting the terms with dy/dx in them on one side and the terms without on the other gives
secē(xy)(dy/dx) - x(dy/dx) = y - secē(xy)y .

Factoring the dy/dx out gives (secē(xy) - x)(dy/dx) = y - secē(xy)y .

Dividing by what is multiplied by the dy/dx gives dy/dx) = (y - secē(xy)y )/(secē(xy) - x).

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