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

Put $s=b^a{t}^{a}x$ and write $y^{\prime }$ for differentiation with respect to $s$. Work on a real branch where the powers are defined and differentiable; $b>0$, $t>0$ suffices for arbitrary real $a$, and the parameter choice below has $b>0$. At fixed $x$, $\partial s/\partial t=as/t$. The product rule and chain rule therefore give

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 $u$.