2.1 Statement

For any real exponent $n \in \mathbb{R}$:

$$ \frac{d}{dx}\bigl[x^n\bigr] = n\,x^{n-1} $$

wherever $x^n$ is defined.

2.2 Proof for Natural Exponents (via the Binomial Theorem)

Let $f(x) = x^n$ where $n \in \mathbb{N}$. By the limit definition:

$$ f'(x) = \lim_{h \to 0} \frac{(x+h)^n - x^n}{h} $$

Step 1 — Expand using the Binomial Theorem (recall this from our earlier topic):

$$ (x+h)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} h^k = x^n + n\,x^{n-1}h + \binom{n}{2}x^{n-2}h^2 + \cdots + h^n $$

Step 2 — Subtract $x^n$ and divide by $h$:

$$ \frac{(x+h)^n - x^n}{h} = \frac{n\,x^{n-1}h + \binom{n}{2}x^{n-2}h^2 + \cdots + h^n}{h} $$ $$ = n\,x^{n-1} + \binom{n}{2}x^{n-2}h + \cdots + h^{n-1} $$

Step 3 — Take the limit as $h \to 0$:

Every term except the first contains a factor of $h$, so they all vanish:

$$ f'(x) = \lim_{h \to 0} \left[ n\,x^{n-1} + \underbrace{\binom{n}{2}x^{n-2}h + \cdots + h^{n-1}}_{\to\, 0} \right] = n\,x^{n-1} \quad \blacksquare $$

2.3 Extension to Negative and Fractional Exponents

Exponent type	Method of extension
Negative integers ($n < 0$)	Write $x^{-m} = \frac{1}{x^m}$ and apply the quotient rule (proved separately).
Rational ($n = p/q$)	Write $y = x^{p/q}$, so $y^q = x^p$. Differentiate implicitly and solve for $y'$.
Arbitrary real $n$	Write $x^n = e^{n \ln x}$ and use the chain rule with exponential/logarithmic derivatives.

2.4 Unification of Special Cases

The power rule is a single formula that unifies several results you may have seen separately: