Chain rule and the derivative of absolute value functions

Question

Is it possible to use the chain rule to calculate the derivative of $|x^4|$ and $|x|^4$ in $x=0$? Does the derivative to these functions exist in $x=0$?

Answer 1

Of course the derivative exists since $|x^4|=x^4$.

But just first ignore the rigor. We have $$\frac{d}{dx}|x^4|=4x^3\frac{d}{dy}|y|=4x^3sgn(y)~~~~~(y=x^4~and~y\neq0)$$ If derivative is computed at $x\neq0$, then $$\frac{d}{dx}|x^4|=4x^3sgn(x^4)=4x^3$$ But for $x=0$, chain rule cannot be applied since $|x|$ doesn't have derivative at $x=0$