In this page we will see how to compute the derivative of the power function using two equivalent forms of the difference quotient: one with the variable \(h\), where \(h\to 0\), and one with the variable \(x\), where \(x\to x_0\).
Let \(n\in\mathbb{N}\setminus\{0\}\), and consider the power function:
\[ f(x)=x^n \]
The two forms of the difference quotient are:
\[ \lim_{h\to 0}\frac{(x+h)^n-x^n}{h} \qquad , \qquad \lim_{x\to x_0}\frac{x^n-x_0^n}{x-x_0} \]
Table of Contents
Difference quotient for \( h\to 0 \)
We compute the derivative of the power function using the definition of the difference quotient:
\[ f'(x) = \lim_{h\to 0} \frac{f(x+h)-f(x)}{h} \]
Substituting \(f(x)=x^n\), we obtain:
\[ f'(x) = \lim_{h\to 0} \frac{(x+h)^n-x^n}{h} \]
We now apply the binomial theorem:
\[ (x+h)^n = x^n + nx^{n-1}h + \frac{n(n-1)}{2}x^{n-2}h^2 + \cdots + h^n \]
Substituting the binomial expansion into the difference quotient:
\[ f'(x) = \lim_{h\to 0} \frac{ x^n + nx^{n-1}h + \frac{n(n-1)}{2}x^{n-2}h^2 + \cdots + h^n - x^n }{h} \]
Simplifying the terms \(x^n\):
\[ f'(x) = \lim_{h\to 0} \frac{ nx^{n-1}h + \frac{n(n-1)}{2}x^{n-2}h^2 + \cdots + h^n }{h} \]
Dividing each term by \(h\):
\[ f'(x) = \lim_{h\to 0} \left( nx^{n-1} + \frac{n(n-1)}{2}x^{n-2}h + \frac{n(n-1)(n-2)}{6}x^{n-3}h^2 + \cdots + h^{n-1} \right) \]
Passing to the limit as \(h\to 0\), all terms containing positive powers of \(h\) tend to \(0\). Therefore:
\[ f'(x) = nx^{n-1} \]
We therefore conclude that:
\[ \frac{d}{dx}x^n = nx^{n-1} \qquad , \qquad \forall x\in\mathbb{R} \]
Difference quotient for \( x\to x_0 \)
Let us now compute the derivative of the power function in the form:
\[ f'(x_0) = \lim_{x\to x_0} \frac{f(x)-f(x_0)}{x-x_0} \]
Substituting \(f(x)=x^n\):
\[ f'(x_0) = \lim_{x\to x_0} \frac{x^n-x_0^n}{x-x_0} \]
The numerator is a difference of powers. We therefore use the factorization:
\[ x^n-x_0^n = (x-x_0) \left( x^{n-1} + x^{n-2}x_0 + \cdots + x_0^{n-1} \right) \]
Substituting into the difference quotient:
\[ f'(x_0) = \lim_{x\to x_0} \frac{ (x-x_0) \left( x^{n-1} + x^{n-2}x_0 + \cdots + x_0^{n-1} \right) }{x-x_0} \]
Simplifying the factor \(x-x_0\):
\[ f'(x_0) = \lim_{x\to x_0} \left( x^{n-1} + x^{n-2}x_0 + \cdots + x_0^{n-1} \right) \]
Passing to the limit as \(x\to x_0\), each term tends to \(x_0^{\,n-1}\). Since there are \(n\) terms equal to \(x_0^{\,n-1}\), we obtain:
\[ f'(x_0) = nx_0^{\,n-1} \]
In conclusion:
\[ f'(x) = nx^{n-1} \qquad , \qquad \forall x\in\mathbb{R} \]