[Math] Cube Root function not differentiable

Question

Why is the cube root function not differentiable at $x=0?$

I graphed it and the curve looks a bit vertical at that point, is that why? Can someone give a good explanation please.

Answer 1

Suppose a function $f(x)=x^n$; its derivative is $f'(x)=nx^{n-1}$. If $n<1$, then rewrite $$f'(x)=nx^{n-1}=\frac{n}{x^{1-n}}$$ the exponent being positive in the denominator, $f'(x)$ is undefined for $x=0$. So, the curve is not a bit vertical.

For you specific case of $f(x)=x^{1/3}$, $f'(x)=\frac{1}{3 x^{2/3}}$, compute the value of $f'(10^{-n})$ for $n=10$ and $n=100$.