Past exam of the year 1 of the computer science course of the University of Oxford / 2013 / Discrete mathematics, probability and continuous mathematics / 4 / b / Solution

The multivariable chain rule follows the change of both coordinates along the parametric curve:Substituting the partial derivatives from part (a), with $x=1+s^2$ and $y=e^s$, yieldsThe first term comes from changing $x$, and the remaining terms from changing $y$.