Partial derivatives are correct. Set them to zero and solve. Btw, notice the exponential term can't be zero, so the other factor is.
Then differentiate again to decide whether your x and y are a min or max, as explained here:
Pauls Online Notes : Calculus III - Relative Minimums and Maximums
f(x,y) = x y e^(y - x^2) - Wolfram|Alpha