Answer
$f_x=2xye^{x^2y}$
$f_y=x^2e^{x^2y}$
Work Step by Step
Take the first partial derivatives of the given function. When taking partial derivative with respect to x, treat y as a constant, and vice versa:
$f_x=e^{x^2y}\times2xy=2xye^{x^2y}$
$f_y=e^{x^2y}\times x^2=x^2e^{x^2y}$