Power Rule for Derivatives

by Justin Skycak (x.com/justinskycak) on January 04, 2019

There are some patterns that allow us to compute derivatives without having to compute the limit of the difference quotient.

This post is part of the book Justin Math: Calculus. Suggested citation: Skycak, J. (2019). Power Rule for Derivatives. In Justin Math: Calculus. https://justinmath.com/power-rule-for-derivatives/

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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.

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

Further Applications

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.

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

Derivation

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

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

Exercises

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

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

Solution:
$\begin{align*} f'(x)= \frac{4}{3} x^\frac{2}{3} \end{align*}$

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

Solution:
$\begin{align*} f'(x)= - \frac{6}{x^7} \end{align*}$

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

Solution:
$\begin{align*} f'(x)= 6\sqrt{x} \end{align*}$

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

Solution:
$\begin{align*} f'(x)=-\frac{2}{5}x^{-\frac{1}{5}} \end{align*}$

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

Solution:
$\begin{align*} f'(x)=-\frac{1}{2}x^{-\frac{3}{2}} \end{align*}$

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

Solution:
$\begin{align*} f'(x)=\frac{1}{72}x^{-\frac{71}{72}} - 2x \end{align*}$

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

Solution:
$\begin{align*} f'(x)= - \frac{15}{2 \sqrt{x^7} } - \frac{3}{2 \sqrt{x}} \end{align*}$

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

Solution:
$\begin{align*} f'(x)= 6.2x^{2.1} + 51x^{101} \end{align*}$

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

Solution:
$\begin{align*} f'(x)= \sqrt{2} x^{\sqrt{2}-1} + 3x^{\sqrt{3}-1} + \frac{2}{ x^{\sqrt{2}+1} } \end{align*}$

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

Solution:
$\begin{align*} -\frac{ \pi e}{x^{\pi+1}} + \pi e x^{e-1} - ex^{ \frac{e}{\pi} - 1 } \end{align*}$

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