On this page we will see how to calculate the derivative of the power function using two equivalent forms to express the difference quotient: for \( h \to 0 \) and for \( x \to x_0 \). Formally, as:
\[ \lim_{h \to 0}\frac{(x + h)^n - x^n}{h} \quad , \quad \lim_{x \to x_0}\frac{x^n - x_0^n}{x - x_0} \]
Table of Contents
- Limit of the difference quotient for \( h \to 0 \)
- Limit of the difference quotient for \( x \to x_0 \)
Limit of the difference quotient for \( h \to 0 \)
We want to calculate the derivative of the function \( f(x) = x^n \) using the definition of the difference quotient:
\[ f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]
Substituting \( f(x) = x^n \):
\[ f'(x) = \lim_{h \to 0} \frac{(x+h)^n - x^n}{h} \]
We use the binomial expansion:
\[ (x+h)^n = x^n + n x^{n-1} h + \frac{n(n-1)}{2} x^{n-2} h^2 + \dots + h^n \]
Substituting:
\[ f'(x) = \lim_{h \to 0} \frac{x^n + n x^{n-1} h + \frac{n(n-1)}{2} x^{n-2} h^2 + \dots + h^n - x^n}{h} \]
Simplifying:
\[ f'(x) = \lim_{h \to 0} \frac{n x^{n-1} h + \frac{n(n-1)}{2} x^{n-2} h^2 + \dots + h^n}{h} \]
Dividing everything by \( h \):
\[ f'(x) = \lim_{h \to 0} \left( n x^{n-1} + \frac{n(n-1)}{2} x^{n-2} h + \frac{n(n-1)(n-2)}{6} x^{n-3} h^2 + \dots + h^{n-1} \right) \]
As \( h \) approaches \( 0 \), all terms containing \( h \) vanish:
\[ f'(x) = n x^{n-1} \]
We therefore conclude that:
\[ \frac{d}{dx} x^n = n x^{n-1}, \quad \forall x \in \mathbb{R} \]
Limit of the difference quotient for \( x \to x_0 \)
Let us calculate the derivative of the power function ( \( f(x) = x^n \) ) as the limit of the difference quotient:
\begin{align} f'(x_0) &= \lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0} \\ &= \lim_{x \to x_0} \frac{x^n - x_0^n}{x - x_0}\end{align}
The numerator of the difference quotient is the difference of powers \( x^n - x_0^n \):
\[ x^n - x_0^n = (x - x_0)(x^{n-1} + x^{n-2} x_0 + \cdots + x_0^{n-1}) \]
Substituting into the expression for the derivative and simplifying:
\begin{align} f'(x_0) &= \lim_{x \to x_0} \frac{(x - x_0)(x^{n-1} + x^{n-2} x_0 + \cdots + x_0^{n-1})}{x - x_0} \\ &= \lim_{x \to x_0} \left(x^{n-1} + x^{n-2} x_0 + \cdots + x_0^{n-1}\right) \end{align}
When \( x \to x_0 \), all terms are evaluated at \( x_0 \):
\[ f'(x_0) = n x_0^{n-1} \]
Therefore, the derivative of the function \( f(x) = x^n \) is:
\[ f'(x) = n x^{n-1} \qquad \forall x \in \mathbb{ R } \]