The Inverse Function Theorem says that if $g(x)$ is differentiable at $a$, and g'(a)\neq 0, then $g^{-1}$ is differentiable at $g(a)$ and (g^{-1})'(g(a)) = \frac{1}{g'(a)}.
Since $f(x) = x^{1/n}$, in order to show that $f(x)$ is differentiable at $a^n$ it is enough to show that $g(x)=x^n$ is differentiable at $a$ and that g'(a)\neq 0, by the Inverse Function Theorem, since $f(x) = g^{-1}(x)$.
Now, if you know that $g(x)=x^n$ is differentiable for all $n$, then you don't need to use induction, just use the Inverse Function Theorem, noting that every value in $(0,\infty)$ is the image of an $a$ in $(0,\infty)$ under $g$, and that g'(a)\neq 0 if $a\neq 0$.
If you don't yet know that $g(x)=x^n$ is differentiable, then you can use induction: $g(x)=x$ is differentiable; if $h(x)=x^n$ is differentiable, then $g(x)=x^{n+1} = h(x)x$, and the product of differentiable functions is differentiable, hence $g(x)$ is differentiable. By induction, $x\mapsto x^n$ is differentiable for all $n\in\mathbb{N}$.