Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2005 / Paper 1 / 11 / c

The chain rule gives $f{f}^{\prime }=\frac{1}{2}(f^2{)}^{\prime }$. Both $f(0)$ and ${\lim }_{x\to \infty }f(x)$ vanish, so the fundamental theorem of calculus givesThe positive contribution while $f$ rises is cancelled by the negative contribution while it falls.