The function $f\left(x\right)=\left|x^3\right|$ is

$$\begin{gathered} Given f\left(x\right)=\left\{\begin{array}{l}-x^3 if x<0\\ 0 if x=0\\ x^3 if x>0\end{array}\right. \\ f\text{'}\left(x\right)=\left\{\begin{array}{l}-3x^{2}if x<0\\ 0 if x=0\\ 3x^2 if x>0\end{array}\right. \\ f\text{'}\left(0^{-}\right)=f^{+}\left(0\right)=0 \\ f is differentiable at x=0 \end{gathered}$$

Hence function is  differentiable every where