Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2009 / Paper 1 / 18 / a / Solution

Regard $u=u(s,t(s,v))$. Applying the multivariable chain rule while holding $v$ fixed givesEach subscript records the variable held fixed. A differentiable local coordinate change from $(s,t)$ to $(s,v)$ requires $(\partial v/\partial t{)}_s\ne 0$; the question's notation $t=t(s,v)$ presupposes such a branch.