Power Rule for Derivatives/Real Number Index/Proof 2

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:


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

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


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

Substituting $x^n$ for $y$:


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

From Exponent Combination Laws:Quotient of Powers:


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