I have an integral to minimize that writes like $$F: \mathbb R^d \to \mathbb R: \theta \mapsto \int_{[0,1]^d} f(\langle x,\theta\rangle) dx$$.
The function $f$ is a convex function, which makes $F$ a convex function.
Q : Let $x \in [0,1]^d$. Is $\frac{df(\langle x,\theta\rangle)}{d\theta}$a subgradient of $F$ at $\theta$ ?