Differentiability and derivative of $|x|^3$ at $x=0$

Question

Let $f(x) = |x|^3$. Using the definition of the derivative, show that $f$ is differentiable at $x = 0$ and find $f'(0)$.

I am attempting to use the formula $\lim_{h\to0}\frac{f(x+h)-f(x)}h$ but I am struggling. Could someone please help?

Answer 1

$\frac{f(0+h)-f(0)}{h}= \frac{|h|^3}{h}=\frac{|h|h^2}{h}=h|h| \to 0$ as $h \to 0$.