Derivative of $F(y)=\int_{0}^{y}\sqrt{x^4-(y-y^2)}dx$?

Question

Let $F(y)=\int_{0}^{y}\sqrt{x^4-(y-y^2)}dx$ for $0 \le y \le 1$. How can one compute the derivative of $F(y)$? I know how to compute such derivative when the integrand is independent of $y$, but I have no idea in this case.

Thank you very much.

Answer 1

Hint: Write $G(y,z)=\int_0^y\sqrt{x^4-(z-z^2)}$ and note that $F(y)=G(y,z(y))$, where $z(y)=y$.