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Calculus/Implicit differentiation

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Expert: - 3/14/2011


Question
Find dy/dx for the function xy+y^2-2x=o

Answer
Think of y as some function of x (which it is), and then see that xy is a product rule
and y² is a power rule.

This gives xy' + y + 2yy' - 2 = 0.
This can be rearranged at xy' + 2yy' = 2 - y.
Factoring out the y' gives y'(x+2y) = 2-y,
Dividing by x+2y gives y' = (2-y)/(x+2y).

That's probably the answer, however ...

The function y could be solved for quadratically from y² + xy - 2x = 0 using y as the variable.
This would give y = (-x ± √(x²+8x))/2.

Putting that in the answer, which is probably not what has to be done, gives
y' = (2-(-x ± √(x²+8x))/2)/(x + 2(-x ± √(x²+8x))/2).  I think that's too messy ...  

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