Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2010 / Paper 1 / 18 / a / ii / Solution

At fixed $t$, the argument is linear in $x$, with $\partial s/\partial x=b^a{t}^a$. Each differentiation therefore contributes this same constant factor, while the outside factor $t^a$ remains unchanged. ThusHere $y^{(3)}$ denotes the third derivative of the single-variable profile, not a partial derivative of $u$.