Your x partial derivative is incorrect. Because x is the lower limit, there's a minus sign associated with it (just think Fundamental Theorem of the Calculus Part II:

\(\displaystyle \displaystyle{\int_{a}^{b}f'(x)\,dx=f(b)-f(a).}\)

To get equations to look like this, you have to use LaTeX. You can find out more about that

here in the stickies.

So, your x partial derivative is

\(\displaystyle \displaystyle{f_{x}(x,y)=-2-e^{x^{2}},}\) and the \(\displaystyle y\) partial is, as you've said,

\(\displaystyle \displaystyle{f_{y}(x,y)=3+e^{y^{2}}.}\)

Query: if you set either of these equal to zero, are there any real solutions? Your "solution" unfortunately has you taking the logarithm of a negative number. Negative numbers are not in the domain of the logarithm function, unless you're thinking a different branch of the complex logarithm function. You're going to run into this problem for both derivatives. What does this tell you?