If $f: \mathbb{R} \to \mathbb{R}$ is weakly differentiable and $u: \Omega \to \mathbb{R}$ is smooth, where $\Omega \subset \mathbb{R}^n$, can we show that for any test function $\varphi \in C_c^1(\Omega)$, we have $$-\int_\Omega f(u(x))\frac{\partial \varphi}{\partial x_i} dx = \int_\Omega f'(u(x))\frac{\partial u}{\partial x_i}\varphi dx$$ where $f'$ is the weak derivative of $f$.
I know that there is a standard result for when $F$ is smooth and has bounded derivative and $v$ is weakly differentiable in $\Omega$, that $F\circ v$ is weakly differentiable, but I am interested in the other way round.
In particular, I am trying to understand Evans and Gariepy page 187.