I know that this is true for all cases, via chain rule:
$$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dv}\frac{dv}{dx}$$
However, is this allowed?
$$(\frac{\partial{y}}{\partial{x}})_z=(\frac{\partial{y}}{\partial{u}})_z(\frac{\partial{u}}{\partial{v}})_z(\frac{\partial{v}}{\partial{x}})_z$$
Assuming the variable that is held constant is z(that does not appear anywhere within the partial derivatives).