$$\frac{\partial \mu(v)}{\partial v}\bigg\vert_{v(\dot{q})} \frac{\partial v}{\partial \dot{q}} \stackrel{{\color{red}?}}{=} \frac{\partial v}{\partial \dot{q}}\frac{\partial \mu(v)}{\partial v}\bigg\vert_{v(\dot{q})}$$
Why is this a valid transformation? I thought the chain rule worked in one direction so to speak:
$$\frac{dy}{dx}\bigg\vert_{x(t)}\frac{dx}{dt} = \frac{dy}{dt}$$