Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2023 / Paper 1 / 14 / c / Solution

Let $t=y/x$. The product rule and the chain rule give the partial derivativesHence the terms containing ${\varphi }^{\prime }$ cancel:This also follows from the Euler theorem for homogeneous functions, because $u(\lambda x,\lambda y)=\lambda u(x,y)$ for $\lambda >0$.