Can $n$ & $a$ be any real number in the power rule?

Question

$$\lim_{x\to a}\frac{x^n-a^n}{x-a}=na^{n-1}\tag{1}$$

$$\frac{d}{da}(a^n)=na^{n-1}\tag{2}$$

In $(1)$ & $(2)$, can $a$ & $n$ be any real number?

Answer 1

The differentiation rule in question applies to all these cases:

1. the base is positive and the exponent is real;
2. the base is real and the exponent is a positive integer;
3. the base is nonzero and the exponent is an integer.

Answer 2

Certainly $a$ cannot be negative if $n$ is irrational or rational with even denominator, since the function $a^n$ is not even defined for negative values of $a$ in those cases. Otherwise the answer is yes; one can prove this for positive $a$ by writing $a^n = e^{n\log a}$ and differentiating using the Chain Rule.