= Solution
Regard $u=u(s,t(s,v))$. Applying the <multivariable chain rule> while holding $v$ fixed gives
$$
\boxed{\left(\frac{\partial u}{\partial s}\right)_v=\left(\frac{\partial u}{\partial s}\right)_t+\left(\frac{\partial u}{\partial t}\right)_s\left(\frac{\partial t}{\partial s}\right)_v.}
$$
Each subscript records the variable held fixed. A differentiable local coordinate change from $(s,t)$ to $(s,v)$ requires $(\partial v/\partial t)_s\ne0$; the question's notation $t=t(s,v)$ presupposes such a branch.
Back to article page
