Consider the real-valued function $f(x)=x^r$ where $r$ is a real number.

(1) For what values of $x$ and $r$, is this function differentiable?

(2) How do you prove the power rule, when $f(x)$ is differentiable?

It would be nice if the proof is accessible to a 12th grade student.

EDIT: Perhaps, I have not been very clear about my question. I am not satisfied with the "textbook proofs" I have seen. Please established the existence (of the derivative) first before proving the power rule. That is, please consider first question first. This is the reason why I call the question a mathematical engineering question. (If we can we bring high level math down to Grade 12 level, without loosing precision, then I call it good mathematical engineering.)