Solution
= Solution
Let $G(s)=\int_0^se^{t^2}\,dt$. The <fundamental theorem of calculus> gives $G'(s)=e^{s^2}$. Applying the <chain rule> to $F(x,y)=G(xy^2)$ gives
$$
\boxed{F_x=y^2e^{x^2y^4},\qquad F_y=2xy\,e^{x^2y^4}.}
$$
No elementary antiderivative of $e^{t^2}$ is needed: differentiation of the variable upper limit supplies the integrand directly.