A real analysis exam paper poses the question:
Prove that $x \to x^p$ has the derivative $px^{p-1}$where $p \in \mathbb R$
My attempt:
Let $f(x) = x^p$
Then the derivative at $c\in \mathbb R$ is the limit
$\lim_{x_\to c}\frac{f(x)-f(c)}{x-c}$ = $\lim_{x_\to c}\frac{x^p-c^p}{x-c}$
Using the binomial expansion(?):
this is equivalent to:
$\lim_{x_\to c}(x^{p-1}+x^{p-2}c+...+c^{p-1})$
Where there are $p$ terms thus the limit becomes
$pc^{p-1}$
thus $\forall x \in \mathbb R$ the derivative is $px^{p-1}$
Any criticism is welcome and needed. Thankyou for reading.