What is f'(0) for this function?

Explanation

1. 01

   Near 0, $x - 1 < 0$, so $\lvert x - 1 \rvert = 1 - x$, with slope $-1$.

2. 02

   $x\lvert x \rvert$ has slope $2\lvert x \rvert = 0$ at 0, so $f'(0) = 0 - 1 = -1$.