Power Rule for Derivatives

Theorem
Let $$n \in \mathbb{R}$$.

Let $$f: \mathbb{R} \to \mathbb{R}$$ be the real function defined as $$f \left({x}\right) = x^n$$.

Then, except for $$n = 0$$ and $$x = 0$$, $$f^{\prime} \left({x}\right) = n x^{n-1}$$.

When $$n = 0$$ and $$x = 0$$, $$f^{\prime} \left({x}\right) = 0$$.

Proof
This can be done in sections.

Proof for Natural Number Index
Let $$f(x)=x^n$$,$$x\in R$$ and $$n\in N$$.

By the definition of the derivative, $$\frac{d}{dx}f(x) = \lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h} = \lim_{h\rightarrow 0} \frac{(x+h)^n-x^n}{h}$$.

Using the binomial theorem this simplifies to:

$$\lim_{h\rightarrow 0}\frac{({n\choose 0}x^n+{n\choose 1}x^{n-1}h+{n\choose 2}x^{n-2}h^2+\dots+{n\choose n-1}xh^{n-1}+{n\choose n}h^n)-x^n}{h}$$

$$=\lim_{h\rightarrow 0}\frac{{n\choose 1}x^{n-1}h+{n\choose 2}x^{n-2}h^2+\dots+{n\choose n-1}xh^{n-1}+{n\choose n}h^n}{h}$$

$$=\lim_{h\rightarrow 0} {n\choose 1}x^{n-1}+{n\choose 2}x^{n-2}h^1+\dots+{n\choose n-1}xh^{n-2}+{n\choose n}h^{n-1}$$

Evaluating the limit gives,

$$={n\choose 1}x^{n-1} = nx^{n-1}$$

Alternative Proof for Natural Number Index
We will use the notation $$D f \left({x}\right) = f^{\prime} \left({x}\right)$$ as it is convenient.

Let $$n = 0$$.

Then $$\forall x \in \mathbb{R}: x^n = 1$$.

Thus $$f \left({x}\right)$$ is the constant function $$f_1 \left({x}\right)$$ on $$\mathbb{R}$$.

Thus from Differentiation of a Constant, $$D f \left({x}\right) = D \left({x^0}\right) = 0 x^{-1}$$, except where $$x = 0$$ in which case $$D \left({1}\right) = 0$$.

So the result holds for $$n = 0$$.

Let $$n = 1$$.

Then $$\forall x \in \mathbb{R}: f \left({x}\right) = x^n = x$$.

Then from Differentiation of the Identity Function $$D \left({x}\right) = 1 = 1 \cdot x^{1-1}$$.

So the result holds for $$n = 1$$.

Now assume $$D \left({x^k}\right) = k x^{k-1}$$.

Then by the Product Rule for Derivatives, $$D \left({x^{k+1}}\right) = D \left({x^k x}\right) = x^k D \left({x}\right) + D \left({x^k}\right) x = x^k \cdot 1 + k x^{k-1} x = \left({k+1}\right) x^k$$.

The result follows by induction.

Proof for Integer Index
When $$n \ge 0$$ we use the result for Natural Number Index.

Now let $$n < 0$$.

Then let $$m = -n$$ and so $$m > 0$$.

Thus $$x^n = \frac 1 x^m$$.

$$ $$ $$ $$

Proof for Real Number Index
We're gonna prove that $$f^{\,'}(x)=nx^{n-1}$$ holds for all $$n$$ real number.

Actually, one has to compute again the limit $$\underset{h\to 0}{\mathop{\lim }}\,\frac{(x+h)^{n}-x^{n}}{h}.$$ Now, let's do some of algebra:

$$\frac{(x+h)^{n}-x^{n}}{h}=\frac{x^{n}}{h}\left\{ \left( 1+\frac{h}{x} \right)^{n}-1 \right\}=\frac{x^{n}}{h}\left\{ \exp \left[ n\ln \left( 1+\frac{h}{x} \right) \right]-1 \right\},$$

hence, $$\frac{(x+h)^{n}-x^{n}}{h}=x^{n}\cdot \frac{\exp \left[ n\ln \left( 1+\frac{h}{x} \right) \right]-1}{n\ln \left( 1+\frac{h}{x} \right)}\cdot \frac{n\ln \left( 1+\frac{h}{x} \right)}{\frac{h}{x}}\cdot \frac{1}{x}\to nx^{n-1}$$ as $$h\to0 .$$

(Note that we used the following facts: $$\underset{x\to 0}{\mathop{\lim }}\,\frac{e^{x}-1}{x}=\underset{x\to 0}{\mathop{\lim }}\,\frac{\ln (1+x)}{x}=1.$$ Following up on this, the above limits are easily calculated by just making the proper change of variables.)