The definition of the derivative practice question 19

Test each claim against the limit that defines $f'(0)$. What does the difference quotient at 0 look like for this $f$?

Compute the difference quotient at 0: $\dfrac{f(0 + h) - f(0)}{h} = \dfrac{h\,|h|}{h}$. What is left after cancelling?

For $x < 0$, $x\,|x| = -x^{2}$; for $x \geq 0$, it is $x^{2}$. Does one formula for $f'$ cover both sides?