Power Rule for Derivatives/Real Number Index/Proof 2

Theorem
Let $n \in \R$.

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

Then:
 * $f^{\prime} \left({x}\right) = n x^{n-1}$

everywhere that $f \left({x}\right) = x^n$ is defined.

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

Proof
Note this proof does not hold for $x = 0$.

Let $y$ = $f \left({x}\right)$.

Then $y = x^n$.

Then:

Using:
 * Derivative of Composite Function
 * Derivative of Constant Multiple
 * Corollary to Primitive of Reciprocal

and taking the derivative of both sides with respect to $x$ gives:


 * $\dfrac 1 y \dfrac {\mathrm d y} {\mathrm d x} = n \dfrac 1 x$

Multiplying both sides of the equation by $y$ yields:


 * $\dfrac {\mathrm d y} {\mathrm d x} = n \dfrac y x$

Substituting $x^n$ for $y$:


 * $\dfrac {\mathrm d y} {\mathrm d x} = n \dfrac {x^n} x$

From Exponent Combination Laws:Quotient of Powers:


 * $\dfrac {\mathrm d y} {\mathrm d x} = n x^{n-1}$