Today I have the following question: Let $\Omega\subset\mathbb R^d$ be open. Say $g:\mathbb R\rightarrow\mathbb R$ is a continuously differentiable function on $\mathbb R$, and $f:\Omega\rightarrow\mathbb R$ is weakly differentiable on $\Omega$. Let $a$ be a real number. Define $F:\Omega\rightarrow\mathbb R$ by: $$ F(x)=\int\limits_a^{f(x)}g(y)\,dy $$
The question: $\textbf{Is $F$ a weakly differentiable function on $\Omega$?}$ Note that, if $f$ is continuously differentiable on $\Omega$, then so is $F$, and by the Fundamental Theorem of Calculus, $$ \nabla F(x)=g(f(x))\nabla f(x). $$ Due to the above we have the expectation that, if $F$ is weakly differentiable, then its weak derivative should be given by the above formula. But this is not clear to me in the general case here described. Any help?