Past exam of the mathematical methods II course of the University of Oxford / 2019 / 5 / Solution

Set $g(x)=\sqrt{8x}+{\sin }^2(2x)$. The chain rule gives $f^{\prime }=-g^{\prime }/g^2$, whileThusThe real domain excludes zero because the original denominator vanishes there; for $x>0$ it is strictly positive.