Power Rule for Derivatives

It can be a pain to evaluate the difference quotient every time we want to take the derivative of a function. Luckily, there are some patterns in derivatives that allow us to compute derivatives without having to go through all the steps of computing the limit of the difference quotient. One such pattern is the power rule, which tells us that the derivative of a function $f(x)=x^n$, where $n$ is some constant number, is given by $f’(x)=nx^{n-1}$. Several examples are shown below.

$\begin{align*} (x^5)'=5x^4 \hspace{1cm} (x) =(x^1)'=1x^0 = 1 \hspace{1cm} (x^{-3})'=-3x^{-4} \end{align*}$

We can also use the power rule to differentiate constants and radical expressions.

$\begin{align*} (1)'=(x^0)'=0x^{-1}=0 \end{align*}$

$\begin{align*} \left( \sqrt{x} \right)' = \left( x^\frac{1}{2} \right)' = \frac{1}{2} x^{-\frac{1}{2}} = \frac{1}{2x^\frac{1}{2}} = \frac{1}{2\sqrt{x}} \end{align*}$

$\begin{align*} \left( \sqrt[3]{x^5} \right)' = \left( x^{\frac{5}{3}} \right)' = \frac{5}{3}x^{\frac{2}{3}} = \frac{5}{3}\sqrt[3]{x^2} \end{align*}$

When a term is multiplied by some constant number, we can move the number outside of the derivative, i.e. we can take the derivative of the term and multiply it by that number.

$\begin{align*} (5)'=(5x^0)'=5(x^0)'=5(0)=0 \end{align*}$

$\begin{align*} \left(−3\sqrt{x} \right)'= −3 \left( \sqrt{x} \right)' = −3 \left( \frac{1}{2\sqrt{x}} \right) = −\frac{3}{2\sqrt{x}} \end{align*}$

$\begin{align*} \left( \frac{1}{4} \sqrt[3]{x^5} \right)' = \frac{1}{4} \left( \sqrt[3]{x^5} \right)' = \frac{1}{4} \left( \frac{5}{3}\sqrt[3]{x^2} \right) = \frac{5}{12}\sqrt[3]{x^2} \end{align*}$

In general, for any number $c$, we have

$\begin{align*} (cx^n)' = c(x^n)'=cnx^{n−1}. \end{align*}$

When we have a sum or difference of terms, we can apply the power rule to each term individually.

$\begin{align*} &\left( \frac{2}{3}x^3 −5 \sqrt{x} + \frac{1}{4x^2} − 7 \right)' \\ &= \left( \frac{2}{3}x^3 \right)' − \left( 5 \sqrt{x} \right)' + \left( \frac{1}{4x^2} \right)' − \left( 7 \right)' \\ &= \frac{2}{3} \left( x^3 \right)' − 5 \left( \sqrt{x} \right)' + \frac{1}{4} \left( \frac{1}{x^2} \right)' − 7 \left( 1 \right)' \\ &= \frac{2}{3} \left( 3x^2 \right) − 5 \left( \frac{1}{2\sqrt{x}} \right) + \frac{1}{4} \left( −\frac{2}{x^3} \right) −7 \left( 0 \right) \\ &= 2x^2 − \frac{5}{2\sqrt{x}} −\frac{1}{2x^3} \end{align*}$

To see why the power rule works, we can compute the derivative for $x^n$ using the difference quotient.

$\begin{align*} (x^n)' &= \lim\limits_{\Delta x \to 0} \frac{(x+\Delta x)^n−x^n}{\Delta x} \\ &= \lim\limits_{\Delta x \to 0} \frac{(x+\Delta x)(x+\Delta x)\cdots (x+\Delta x) −x^n}{\Delta x} \\ &= \lim\limits_{\Delta x \to 0} \frac{x^n + nx^{n−1} \Delta x + \left[ \mbox{other terms of at least } (\Delta x)^2 \right] −x^n}{\Delta x} \\ &= \lim\limits_{\Delta x \to 0} \frac{nx^{n−1} \Delta x + \left[ \mbox{other terms of at least } (\Delta x)^2 \right] }{\Delta x} \\ &= \lim\limits_{\Delta x \to 0} \frac{nx^{n−1} \Delta x + (\Delta x)^2(\mbox{other terms})}{\Delta x} \\ &= \lim\limits_{\Delta x \to 0} \left[ nx^{n−1} + (\Delta x) (\mbox{other terms}) \right] \\ &= nx^{n−1} + (0) (\mbox{other terms}) \\ &= nx^{n−1} + 0 \\ &= nx^{n−1} \end{align*}$

Practice Problems

Use the power rule to differentiate the following functions. (You can view the solution by clicking on the problem.)

$\begin{align*}1) \hspace{.5cm} f(x)=x^\frac{4}{3} \end{align*}$
$\begin{align*} f'(x)= \frac{4}{3} x^\frac{2}{3} \end{align*}$

$\begin{align*}2) \hspace{.5cm} f(x)=\frac{1}{x^6}\end{align*}$
$\begin{align*} f'(x)= - \frac{6}{x^7} \end{align*}$

$\begin{align*}3) \hspace{.5cm} f(x)=4 \sqrt{x^3} \end{align*}$
$\begin{align*} f'(x)= 6\sqrt{x} \end{align*}$

$\begin{align*}4) \hspace{.5cm} f(x)=-\frac{1}{2}x^\frac{4}{5} \end{align*}$
$\begin{align*} f'(x)=-\frac{2}{5}x^{-\frac{1}{5}} \end{align*}$

$\begin{align*}5) \hspace{.5cm} f(x)=\frac{1}{\sqrt{x}}+3 \end{align*}$
$\begin{align*} f'(x)=-\frac{1}{2}x^{-\frac{3}{2}} \end{align*}$

$\begin{align*}6) \hspace{.5cm} f(x)=x^\frac{1}{72}-x^2 \end{align*}$
$\begin{align*} f'(x)=\frac{1}{72}x^{-\frac{71}{72}} - 2x \end{align*}$

$\begin{align*}7) \hspace{.5cm} f(x)=\frac{3}{\sqrt{x^5}}-3\sqrt{x} \end{align*}$
$\begin{align*} f'(x)= - \frac{15}{2 \sqrt{x^7} } - \frac{3}{2 \sqrt{x}} \end{align*}$

$\begin{align*}8)\hspace{.5cm} f(x) = 3x^{3.1} + \frac{1}{2} x^{102} \end{align*}$
$\begin{align*} f'(x)= 6.2x^{2.1} + 51x^{101} \end{align*}$

$\begin{align*}9) \hspace{.5cm} f(x) = x^\sqrt{2} + \sqrt{3} x^\sqrt{3} - \frac{ \sqrt{2} }{x^\sqrt{2} } \end{align*}$
$\begin{align*} f'(x)= \sqrt{2} x^{\sqrt{2}-1} + 3x^{\sqrt{3}-1} + \frac{2}{ x^{\sqrt{2}+1} } \end{align*}$

$\begin{align*}10) \hspace{.5cm} f(x) = \frac{3}{x^\pi} + \pi x^e - \pi x^\frac{e}{\pi} \end{align*}$
$\begin{align*} -\frac{ \pi e}{x^{\pi+1}} + \pi e x^{e-1} - ex^{ \frac{e}{\pi} - 1 } \end{align*}$

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