Second-order partial derivative problem

Question

I need to find the second-order partial derivative $\frac{\partial^2z}{\partial x^2}$ of the following problem:

$$z(x,y) = \frac{1}{2}\ln(x²+y²)$$

I got the first part:

First derivative is $$\frac{x}{x² + y²}$$

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I then thought I would use the quotient rule to find the second derivative of x.

So this would mean: $$\frac{(x²+y²)-2x²}{(x²+y²)²}$$ right? But the correct answer is: $\dfrac{y² - x²}{(x²+y²)²}$

Anyone care to explain why and what I did wrong?

Answer 1

Community wiki answer so the question can be marked as answered:

As Piwi noted in a comment, your result is correct and you merely failed to simplify the numerator.