Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 1 / 10E / b / ii / Solution

No. Take $a=0$ andThen $f$ is differentiable andSince $x|x|/x=|x|\to 0$, the product is differentiable at zero withHowever, $g$ has one-sided derivatives $-1$ and $1$ and is not differentiable at zero. This is a nondifferentiable factor with a differentiable nondegenerate product.