Derivatives of Power Functions

One of the most basic types of differentiable functions are power functions. The following theorem will give us a method to differentiate power functions. The proof is rather cumbersome, so the reader may skip it at their own desire.

Theorem 1: If $f$ is a power function, that is $f(x) = x^n$ where $n \in \mathbb{Z}^+$, then the derivative of $f$ can be obtained from the formula $f'(x) = \frac{d}{dx} x^n = nx^{n-1}$.

• Partial Proof: We will prove this property for $n \in \mathbb{N}$ using the binomial theorem which says that $(x + y)^n = \begin{pmatrix} n\\ 0 \end{pmatrix} x^ny^0 + \begin{pmatrix} n\\ 1 \end{pmatrix} x^{n-1}y^1 + ... + \begin{pmatrix} n\\ n-1 \end{pmatrix} x^1y^{n-1} + \begin{pmatrix} n\\ n \end{pmatrix} x^0y^n$, where $\begin{pmatrix} n \\ r \end{pmatrix} = \frac{n!}{r!(n-r)!}$ and where $n! = n \cdot (n-1) \cdot ... \cdot 2 \cdot 1$.

• Suppose that $f(x) = x^n$. Applying the definition of the derivative, we get that:

(1)

\begin{align} f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \\ f'(x) = \lim_{h \to 0} \frac{(x + h)^n - x^n}{h} \\ f'(x) = \lim_{h \to 0} \frac{\begin{pmatrix} n\\ 0 \end{pmatrix} x^n + \begin{pmatrix} n\\ 1 \end{pmatrix}x^{n-1}h + \begin{pmatrix} n\\ 2 \end{pmatrix}x^{n-2}h^2 + ... + \begin{pmatrix} n\\ n-1 \end{pmatrix} xh^{n-1} + \begin{pmatrix} n\\ n \end{pmatrix} h^n - x^n}{h} \\ f'(x) = \lim_{h \to 0} \frac{x^n + \begin{pmatrix} n\\ 1 \end{pmatrix}x^{n-1}h + \begin{pmatrix} n\\ 2 \end{pmatrix}x^{n-2}h^2 + ... + \begin{pmatrix} n\\ n-1 \end{pmatrix} xh^{n-1} + h^n - x^n}{h} \\ f'(x) = \lim_{h \to 0} \frac{\begin{pmatrix} n\\ 1 \end{pmatrix}x^{n-1}h + \begin{pmatrix} n\\ 2 \end{pmatrix}x^{n-2}h^2 + ... + \begin{pmatrix} n\\ n-1 \end{pmatrix} xh^{n-1} + h^n }{h} \\ f'(x) = \lim_{h \to 0} \begin{pmatrix} n\\ 1 \end{pmatrix}x^{n-1} + \begin{pmatrix} n\\ 2 \end{pmatrix}x^{n-2}h^1 + ... + \begin{pmatrix} n\\ n-1 \end{pmatrix} xh^{n-2} + h^{n-1} \\ f'(x) = \lim_{h \to 0} \begin{pmatrix} n\\ 1 \end{pmatrix}x^{n-1} + \lim_{x \to 0} \begin{pmatrix} n\\ 2 \end{pmatrix}x^{n-2}h^1 + ... + \begin{pmatrix} n\\ n-1 \end{pmatrix} xh^{n-2} + h^{n-1} \\ f'(x) = \lim_{h \to 0} \begin{pmatrix} n\\ 1 \end{pmatrix}x^{n-1} + 0 \\ f'(x) = \lim_{h \to 0} nx^{n-1} = nx^{n-1} \quad \blacksquare \end{align}

We can now look at some examples of applying this rule without going through the tedium of using the definition of a derivative.

Example 1

Differentiate the following function $f(x) = -3x^{12}$.

Applying the property above, we get that:

(2)

\begin{align} \quad \frac{d}{dx} -3x^{12} = -3\frac{d}{dx} x^{12} = -3 \cdot 12x^{11} = -36x^{11} \end{align}

Example 2

Differentiate the function $f(x) = 3x^2 + 4x^3 + 5x^4$.

One again, applying the property at the top of the page and other derivative rules we get:

(3)

\begin{align} \quad \frac{d}{dx} ( 3x^2 + 4x^3 + 5x^4 ) = 6x + 12x^2 + 20x^3 \end{align}

Example 3

Differentiate the function $g(x) = \sqrt{x}$.

You may think you cannot evaluate this derivative with the power rule, though, recall that radicals can be written as exponents such that:

(4)

\begin{equation} g(x) = x^{0.5} \end{equation}

This can now be evaluated such that:

(5)

\begin{align} \frac{d}{dx} \sqrt{x} = \frac{1}{2} x^{-1/2} \end{align}