The following is Exercise VII.5.15.(iii) from Analysis II by Amann & Escher.
Suppose $X$ is open in $\mathbb{R}^m$. For $f,g\in C^1(X,\mathbb{R}^m)$, define $[f,g]\in C(X,\mathbb{R}^m)$ by $$[f,g](x):=\partial f(x)g(x)−\partial g(x)f(x)$$ for $x\in X$. We call $[f,g]$ the Lie bracket of $f$ and $g$.
(iii) $[\varphi f,\psi g]=\varphi\psi[f,g]+(\nabla\varphi|f)\psi f−(\nabla\psi|g)\varphi g$ for $\varphi,\psi\in C^1(X,\mathbb{R})$.
After some routine calculations I got $$[\varphi f,\psi g]=\varphi\psi[f,g]+(\nabla\varphi|g)\psi f−(\nabla\psi|f)\varphi g.$$ Am I wrong or is this a typo?
Given that the calculations are routine and a bit tedious, I will not include my attempt here... I hope this doesn't make the question ill-posed.