Let $U \subset \mathbb{R}^n$ be bounded. If $f, f^2, \nabla f^2 \in L^2(U)$, Is $\nabla f$ in $L^2(U)$.
I see that
$$\|\nabla f^2 \|^2_{L^2(U)} = 2\|f \cdot \nabla f \|^2_{L^2(U)} \leq \|f^2 \|^2_{L^2(U)} + \|[\nabla f]^2 \|^2_{L^2(U)}$$
by Young's inequality which does not take me anywhere. Any hint?