# Rules of Differentiation for Finding Derivatives

The good news is that you don’t have to differentiate all expressions with the definition of the derivative. Here are all the rules you can follow to be able to differentiate the basic functions.

Look closely at the examples, and make sure you understand what is happening.

Rule

### RulesforDerivation

$\begin{array}{llllllllllll}\hfill f\left(x\right)& =k\phantom{\rule{2em}{0ex}}& \hfill ⇔& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& =0\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill f\left(x\right)& =ax\phantom{\rule{2em}{0ex}}& \hfill ⇒& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& =a\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill f\left(x\right)& ={x}^{n}\phantom{\rule{2em}{0ex}}& \hfill ⇒& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& =n{x}^{n-1}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill f\left(x\right)& =\sqrt{x}={x}^{\frac{1}{2}}\phantom{\rule{2em}{0ex}}& \hfill ⇒& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& =\frac{1}{2}{x}^{-\frac{1}{2}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & =\frac{1}{2\sqrt{x}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill f\left(x\right)& ={e}^{x}\phantom{\rule{2em}{0ex}}& \hfill ⇒& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& ={e}^{x}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill f\left(x\right)& ={e}^{kx}\phantom{\rule{2em}{0ex}}& \hfill ⇒& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& =k{e}^{kx}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill f\left(x\right)& ={a}^{x}\phantom{\rule{2em}{0ex}}& \hfill ⇒& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& ={a}^{x}\mathrm{ln}a\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill f\left(x\right)& =\mathrm{ln}x\phantom{\rule{2em}{0ex}}& \hfill ⇒& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& =\frac{1}{x}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill f\left(x\right)& =\mathrm{ln}\left(kx\right)\phantom{\rule{2em}{0ex}}& \hfill ⇒& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& =\frac{1}{x}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill f\left(x\right)& =\mathrm{sin}x\phantom{\rule{2em}{0ex}}& \hfill ⇒& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& =\mathrm{cos}x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill f\left(x\right)& =\mathrm{cos}x\phantom{\rule{2em}{0ex}}& \hfill ⇒& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& =-\mathrm{sin}x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill f\left(x\right)& =\mathrm{sin}\left(kx\right)\phantom{\rule{2em}{0ex}}& \hfill ⇒& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& =k\mathrm{cos}\left(kx\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill f\left(x\right)& =\mathrm{cos}\left(kx\right)\phantom{\rule{2em}{0ex}}& \hfill ⇒& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& =-k\mathrm{sin}\left(kx\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill f\left(x\right)& =\mathrm{tan}x\phantom{\rule{2em}{0ex}}& \hfill ⇒& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& =\frac{1}{{\mathrm{cos}}^{2}x}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & ={\mathrm{tan}}^{2}x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill f\left(x\right)& =\mathrm{tan}\left(kx\right)\phantom{\rule{2em}{0ex}}& \hfill ⇒& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& =\frac{k}{{\mathrm{cos}}^{2}\left(kx\right)}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & =k+k{\mathrm{tan}}^{2}\left(kx\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

$\begin{array}{llllllllllll}\hfill f\left(x\right)& =k\phantom{\rule{2em}{0ex}}& \hfill ⇔& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& =0\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill f\left(x\right)& =ax\phantom{\rule{2em}{0ex}}& \hfill ⇒& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& =a\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill f\left(x\right)& ={x}^{n}\phantom{\rule{2em}{0ex}}& \hfill ⇒& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& =n{x}^{n-1}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill f\left(x\right)& =\sqrt{x}={x}^{\frac{1}{2}}\phantom{\rule{2em}{0ex}}& \hfill ⇒& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& =\frac{1}{2}{x}^{-\frac{1}{2}}=\frac{1}{2\sqrt{x}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill f\left(x\right)& ={e}^{x}\phantom{\rule{2em}{0ex}}& \hfill ⇒& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& ={e}^{x}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill f\left(x\right)& ={e}^{kx}\phantom{\rule{2em}{0ex}}& \hfill ⇒& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& =k{e}^{kx}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill f\left(x\right)& ={a}^{x}\phantom{\rule{2em}{0ex}}& \hfill ⇒& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& ={a}^{x}\mathrm{ln}a\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill f\left(x\right)& =\mathrm{ln}x\phantom{\rule{2em}{0ex}}& \hfill ⇒& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& =\frac{1}{x}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill f\left(x\right)& =\mathrm{ln}\left(kx\right)\phantom{\rule{2em}{0ex}}& \hfill ⇒& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& =\frac{1}{x}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill f\left(x\right)& =\mathrm{sin}x\phantom{\rule{2em}{0ex}}& \hfill ⇒& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& =\mathrm{cos}x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill f\left(x\right)& =\mathrm{cos}x\phantom{\rule{2em}{0ex}}& \hfill ⇒& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& =-\mathrm{sin}x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill f\left(x\right)& =\mathrm{sin}\left(kx\right)\phantom{\rule{2em}{0ex}}& \hfill ⇒& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& =k\mathrm{cos}\left(kx\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill f\left(x\right)& =\mathrm{cos}\left(kx\right)\phantom{\rule{2em}{0ex}}& \hfill ⇒& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& =-k\mathrm{sin}\left(kx\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill f\left(x\right)& =\mathrm{tan}x\phantom{\rule{2em}{0ex}}& \hfill ⇒& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& =\frac{1}{{\mathrm{cos}}^{2}x}={\mathrm{tan}}^{2}x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill f\left(x\right)& =\mathrm{tan}\left(kx\right)\phantom{\rule{2em}{0ex}}& \hfill ⇒& \phantom{\rule{2em}{0ex}}& \hfill {f}^{\prime }\left(x\right)& =\frac{k}{{\mathrm{cos}}^{2}\left(kx\right)}=k+k{\mathrm{tan}}^{2}\left(kx\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

For functions $f\left(x\right)$, $g\left(x\right)$ and constants $k$, the following applies:

$\begin{array}{llll}\hfill {\left(f\left(x\right)±g\left(x\right)\right)}^{\prime }& ={f}^{\prime }\left(x\right)±{g}^{\prime }\left(x\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill {\left(kf\left(x\right)\right)}^{\prime }& =k{f}^{\prime }\left(x\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

In addition to these rules, you can read the entries about the chain rule, the product rule and the quotient rule.

Example 1

 $f\left(x\right)=6⇒{f}^{\prime }\left(x\right)=0$

Example 2

 $f\left(x\right)=12x⇒{f}^{\prime }\left(x\right)=12$

Example 3

 $f\left(x\right)={x}^{3}⇒{f}^{\prime }\left(x\right)=3{x}^{2}$

Example 4

 $f\left(x\right)=7{x}^{5}⇒{f}^{\prime }\left(x\right)=35{x}^{4}$

Example 5

 $f\left(x\right)=2{x}^{3}-6{x}^{9}⇒{f}^{\prime }\left(x\right)=6{x}^{2}-54{x}^{8}$

 $f\left(x\right)=2{x}^{3}-6{x}^{9}⇒{f}^{\prime }\left(x\right)=6{x}^{2}-54{x}^{8}$

Example 6

 $f\left(x\right)=\frac{1}{x}={x}^{-1}⇒{f}^{\prime }\left(x\right)=-\frac{1}{{x}^{2}}$

Example 7

 $f\left(x\right)={e}^{4x}⇒{f}^{\prime }\left(x\right)=4{e}^{4x}$

Example 8

 $f\left(x\right)={3}^{x}⇒{f}^{\prime }\left(x\right)={3}^{x}\mathrm{ln}3$

Example 9

 $f\left(x\right)=\mathrm{ln}4x⇒{f}^{\prime }\left(x\right)=\frac{1}{x}$

Example 10

Jax Teller is driving along a straight country road in Italy. The number $s$ of kilometers Jax has covered after $t$ hours is given by the function $s\left(t\right)=50{t}^{2}$. What can you say about his speed?

You find Jax’s speed after $t$ hours by differentiating the function $s\left(t\right)$:

 ${s}^{\prime }\left(t\right)=50\cdot 2t=100t$

According to the model, Jax increases his speed by $100$ km/h every hour.

After 30 minutes ($0.5$ hours), his speed is

 ${s}^{\prime }\left(0.5\right)=\left(100\cdot 0.5\right)\phantom{\rule{0.33em}{0ex}}\text{km/h}=50\phantom{\rule{0.17em}{0ex}}\text{km/h}$

After 1 hour, his speed is

 ${s}^{\prime }\left(1\right)=\left(100\cdot 1\right)\phantom{\rule{0.33em}{0ex}}\text{km/h}=100\phantom{\rule{0.17em}{0ex}}\text{km/h}$

After $2.5$ hours, his speed is

 ${s}^{\prime }\left(2.5\right)=\left(100\cdot 2.5\right)\phantom{\rule{0.33em}{0ex}}\text{km/h}=250\phantom{\rule{0.17em}{0ex}}\text{km/h}$

When you drive a long time in a straight direction, you can start experiencing speed blindness, which means you don’t notice that you’re driving too fast!