proof of differentiatiable function

Question

prove that x^(1/3) is differentiable at a with f(a)'=((a^(1/3))^-2)/3 for all a not equal to 0.

I tried a epsilon-delta proof with limes theorem, and or that does not work or I am making somewhere mistake, if any one can help I would really appreciated it!Thank you!

Answer 1

Hint: You can use:

$$\frac{x^{\frac 13} - a^{\frac 13}}{x-a}=\frac 1{x^{\frac 23} +(xa)^{\frac 13} + a^{\frac 23}} $$

and calculate the limite : $$f'(a)=\lim_{x \to a} \frac{x^{\frac 13} - a^{\frac 13}}{x-a}=\lim_{x \to a} \left(\frac 1{x^{\frac 23} +(xa)^{\frac 13} + a^{\frac 23}} \right)$$

Answer 2

As $f$ is the inverse of the differentiable function $g\colon \mathbb R\to\mathbb R$, defined by $x\mapsto x^3$, we know that $f$ is differentiable in $y=g(x)$ wherever $g'(x)\neq0$