# Need help finding the derivative of y = ln(x^2 + y^2)

#### ktehspynx

I know I need to use implicit differentiation.... but I can't seem to solve for y' when y = ln(x^2 + y^2) Please help me, I'd really appreciate it!

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#### drumist

I know I need to use implicit differentiation.... but I can't seem to solve for y' Please help me, I'd really appreciate it!
One method you can use is to first get rid of the logarithm:

$$\displaystyle y=\ln(x^2+y^2) \implies e^y = x^2+y^2$$

This is somewhat easier to differentiate, but the original way works too:

$$\displaystyle y = \ln(x^2 + y^2)$$

$$\displaystyle \implies y' = \frac{\frac{d}{dx}[x^2+y^2]}{x^2+y^2} = \frac{ 2x + 2y y' }{x^2+y^2}$$

To solve for $$\displaystyle y'$$ we need to multiply the denominator over first:

$$\displaystyle \implies (x^2+y^2)y' = 2x + 2y y'$$

$$\displaystyle \implies (x^2+y^2-2y)y' = 2x$$

$$\displaystyle \implies y' = \frac{2x}{x^2+y^2-2y}$$

#### ktehspynx

Thanks so much, this helped a lot. 