I am trying to guess when a function is differentiable: I need this in order to prove that using the implicit and logarithmic differentiation method we can prove that
$$
\dfrac{\operatorname{d\!}}{\operatorname{d\!}x} x^n = n x^{n-1}.
$$
In order to prove that equation we should apply implicit differentiation and thus we need to know that this implicit function is differentiable.
In this intance we have to prove that the $x^n$ where $n$ is any real number is differentiatable without show what it's derivative is.
While working on this problem I think I have found a general theorem which allows to determine if a function is differentiable without actually calculating its derivative, and I need your help for proving it.
Statement of the to be proved theorem:
If for a given function $f:A\to\Bbb R$ at a given point $a\in A$ the set
$$
S=\left\{\dfrac{f(x)-f(a)}{x-a}\; \bigg|\; a \leq x \leq a+h\right\}
$$ is such that there exist finite both $\sup(S)$ and $\inf(S)$, then the $f(x)$ is differentiable in $a$.
I found the statement thanks to an intuition, but I am not able to give a formal proof this Theorem. As a bonus question, I'd like to know how can I show, by using the above statement, that the derivative of $x^n$ exists.