### 1 Table of derivatives

Table 1 lists some of the common functions used in engineering and their corresponding derivatives. Remember that in each case the function in the right-hand column gives the rate of change, or the gradient of the graph, of the function on the left at a particular value of $x$ .

N.B. The angle must always be in radians when differentiating trigonometric functions.

Table 1

Common functions and their derivatives
(In this table $k$ , $n$ and $c$ are constants)

 Function Derivative constant 0 $x$ 1 $kx$ $k$ ${x}^{n}$ $n{x}^{n-1}$ $k{x}^{n}$ $kn{x}^{n-1}$ ${e}^{x}$ ${e}^{x}$ ${e}^{kx}$ $k{e}^{kx}$ $lnx$ $1∕x$ $lnkx$ $1∕x$ $sinx$ $cosx$ $sinkx$ $kcoskx$ $sin\left(kx+c\right)$ $kcos\left(kx+c\right)$ $cosx$ $-sinx$ $coskx$ $-ksinkx$ $cos\left(kx+c\right)$ $-ksin\left(kx+c\right)$ $tanx$ ${sec}^{2}x$ $tankx$ $k{sec}^{2}kx$ $tan\left(kx+c\right)$ $k{sec}^{2}\left(kx+c\right)$

In the trigonometric functions the angle is in radians.

##### Key Point 4

Particularly important is the rule for differentiating powers of functions:

$\text{If}\phantom{\rule{1em}{0ex}}y={x}^{n}\phantom{\rule{1em}{0ex}}\text{then}\phantom{\rule{2em}{0ex}}\frac{dy}{dx}=n{x}^{n-1}$

For example, if $y={x}^{3}$ then $\frac{dy}{dx}=3{x}^{2}$ .

##### Example 2

Use Table 1 to find $\frac{dy}{dx}$ when $y$ is given by

1.   $7x$
2. 14
3. $5{x}^{2}$
4.   $4{x}^{7}$
##### Solution
1. We note that $7x$ is of the form $kx$ where $k=7$ . Using Table 1 we then have $\frac{dy}{dx}=7$ .
2. Noting that 14 is a constant we see that $\frac{dy}{dx}=0$ .
3. We see that $5{x}^{2}$ is of the form $k{x}^{n}$ , with $k=5$ and $n=2$ . The derivative, $kn{x}^{n-1}$ , is then

$10{x}^{1}$ , or more simply, $10x$ . So if $y=5{x}^{2}$ , then $\frac{dy}{dx}=10x$ .

4. We see that $4{x}^{7}$ is of the form $k{x}^{n}$ , with $k=4$ and $n=7$ . Hence the derivative, $\frac{dy}{dx}$ , is

given by $28{x}^{6}$ .

Use Table 1 to find $\frac{dy}{dx}$ when $y$ is

1.   $\sqrt{x}$
2.   $\frac{5}{{x}^{3}}$
1. Write $\sqrt{x}$ as ${x}^{\frac{1}{2}}$ , and use the result for differentiating ${x}^{n}$ with $n=\frac{1}{2}$ .

$\frac{dy}{dx}=n{x}^{n-1}=\frac{1}{2}{x}^{\frac{1}{2}-1}=\frac{1}{2}{x}^{-\frac{1}{2}}$ . This may be written as $\frac{1}{2\sqrt{x}}$ .

2. Write $\frac{5}{{x}^{3}}$ as $5{x}^{-3}$ and use the result for differentiating $k{x}^{n}$ with $k=5$ and $n=-3$ .

$5\left(-3\right){x}^{-3-1}=-15{x}^{-4}$ Although Table 1 is written using $x$ as the independent variable, the Table can be used for any variable.

Use Table 1 to find

1. $\frac{dz}{dt}$ given  $z={e}^{t}$
2. $\frac{dp}{dt}$ given $p={e}^{8t}$
3. $\frac{dz}{dy}$ given $z={e}^{-3y}$
1. From Table 1, if $y={e}^{x}$ , then $\frac{dy}{dx}={e}^{x}$ . Hence if $z={e}^{t}$ then $\frac{dz}{dt}={e}^{t}$ .

2. $8{e}^{8t}$

3. $-3{e}^{-3y}$

Find the derivative, $\frac{dy}{dx}$ , when $y$ is

1. $sin2x$
2. $cos\frac{x}{2}$
3. $tan5x$
1. Use the result for $sinkx$ in Table 1, taking $k=2$ :

$2cos2x$

2. Note that $cos\frac{x}{2}$ is the same as $cos\frac{1}{2}x$ . Use the result for $coskx$ in Table 1:

$-\frac{1}{2}sin\frac{x}{2}$

3. Use the result for $tankx$ in Table 1:

$5{sec}^{2}5x$

##### Exercises
1. Find the derivatives of the following functions with respect to $x$ :
1. $9{x}^{2}$
2. $5$
3. $6{x}^{3}$
4. $-13{x}^{4}$
2. Find $\frac{dz}{dt}$ when $z$ is given by:
1. $\frac{5}{{t}^{3}}$
2. $\sqrt{{t}^{3}}$
3. $5{t}^{-2}$
4. $-\frac{3}{2}{t}^{\frac{3}{2}}$
5. $ln5t$
3. Find the derivative of each of the following with respect to the appropriate variable:
1. $sin5x$
2. $cos4t$
3. $tan3r$
4. ${e}^{2v}$
5. $\frac{1}{{e}^{3t}}$
4. Find the derivatives of the following with respect to $x$ :
1. $cos\frac{2x}{3}$
2. $sin\left(-2x\right)$
3. $tan\pi x$
4. ${e}^{\frac{x}{2}}$
5. $ln\frac{2}{3}x$
1. $18x$
2. 0
3. $18{x}^{2}$
4. $-52{x}^{3}$
1. $-15{t}^{-4}$
2. $\frac{3}{2}{t}^{\frac{1}{2}}$
3. $-10{t}^{-3}$
4. $-\frac{9}{4}{t}^{\frac{1}{2}}$
5. $\frac{1}{t}$
1. $5cos5x$
2. $-4sin4t$
3. $3{sec}^{2}3r$
4. $2{e}^{2v}$
5. $-3{e}^{-3t}$
1. $-\frac{2}{3}sin\frac{2x}{3}$
2. $-2cos\left(-2x\right)$
3. $\pi {sec}^{2}\pi x$
4. $\frac{1}{2}{e}^{\frac{x}{2}}$
5. $\frac{1}{x}$