### 2 Notation for derivatives

Just as there is a notation for the first derivative so there is a similar notation for higher derivatives. Consider the function, $y\left(x\right)$ . We know that the first derivative is $\frac{dy}{dx}$ or $\frac{d}{dx}\left(y\right)$ which is the instruction to differentiate the function $y\left(x\right)$ . The second derivative is calculated by differentiating the first derivative, that is

So, using a fairly obvious adaptation of our derivative notation, the second derivative is denoted by $\frac{{d}^{2}y}{d{x}^{2}}$ and is read as ‘dee two $y$ by dee $x$ squared’. This is often written more concisely as ${y}^{″}$ .

In similar manner, the third derivative is denoted by $\frac{{d}^{3}y}{d{x}^{3}}$ or ${y}^{‴}$ and so on. So, referring to Example 6 we could have written

##### Key Point 7

If $y=y\left(x\right)$ then its first, second and third derivatives are denoted by:

$\frac{dy}{dx}\phantom{\rule{2em}{0ex}}\frac{{d}^{2}y}{d{x}^{2}}\phantom{\rule{2em}{0ex}}\frac{{d}^{3}y}{d{x}^{3}}$
$\phantom{\rule{1em}{0ex}}\text{or}\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}\phantom{\rule{2em}{0ex}}{y}^{\prime }\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}{y}^{″}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}{y}^{‴}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}$

In most examples we use $x$ to denote the independent variable and $y$ the dependent variable. However, in many applications, time $t$ is the independent variable. In this case a special notation is used for derivatives. Derivatives with respect to $t$ are often indicated using a dot notation, so $\frac{dy}{dt}$ can be written as $ẏ$ , pronounced ‘ $y$ dot’. Similarly, a second derivative with respect to $t$ can be written as $ÿ$ , pronounced ‘ $y$ double dot’.

##### Key Point 8

If $y=y\left(t\right)$ then

Calculate $\frac{{d}^{2}y}{d{t}^{2}}$ and $\frac{{d}^{3}y}{d{t}^{3}}$ given $y={e}^{2t}+cost$ .

First find $\frac{dy}{dx}$ :

$\frac{dy}{dt}=2{e}^{2t}-sint$

Now obtain the second derivative:

$4{e}^{2t}-cost$

Finally, obtain the third derivative:

$8{e}^{2t}+sint$

Note that in the last Task we could have used the dot notation and written $ẏ=2{e}^{2t}-sint$ , $ÿ=4{e}^{2t}-cost$ and $ÿ\phantom{\rule{0.3em}{0ex}}\stackrel{̇}{}=8{\text{e}}^{2t}+sint$

We may need to evaluate higher derivatives at specific points. We use an obvious notation.

The second derivative of $y\left(x\right)$ , evaluated at say, $x=2$ , is written as $\frac{{d}^{2}y}{d{x}^{2}}\left(2\right)$ , or more simply as ${y}^{″}\left(2\right)$ . The third derivative evaluated at $x=-1$ is written as $\frac{{d}^{3}y}{d{x}^{3}}\left(-1\right)$ or ${y}^{‴}\left(-1\right)$ .

Given $y\left(x\right)=2sinx+3{x}^{2}$ find

1.   ${y}^{\prime }\left(1\right)$
2. ${y}^{″}\left(-1\right)$
3. ${y}^{‴}\left(0\right)$

First find ${y}^{\prime }\left(x\right),{y}^{″}\left(x\right)$ and ${y}^{‴}\left(x\right)$ :

${y}^{\prime }\left(x\right)=2cosx+6x\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}{y}^{″}\left(x\right)=-2sinx+6\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}{y}^{‴}\left(x\right)=-2cosx$

Now substitute $x=1$ in ${y}^{\prime }\left(x\right)$ to obtain ${y}^{\prime }\left(1\right)$ :

${y}^{\prime }\left(1\right)=2cos1+6\left(1\right)=7.0806$ . Remember, in $cos1$ the ‘ $1$ ’ is $1$ radian.

Now find ${y}^{″}\left(-1\right)$ :

${y}^{″}\left(-1\right)=-2sin\left(-1\right)+6=7.6829$ Finally, find ${y}^{‴}\left(0\right)$ :

${y}^{‴}\left(0\right)=-2cos0=-2$ .

##### Exercises
1. Find $\frac{{d}^{2}y}{d{x}^{2}}$ where $y\left(x\right)$ is defined by:
1. $3{x}^{2}-{e}^{2x}$
2. $sin3x+cosx$
3.   $\sqrt{x}$
4. ${e}^{x}+{e}^{-x}$
5. $1+x+{x}^{2}+lnx$
2. Find $\frac{{d}^{3}y}{d{x}^{3}}$ where $y$ is given in Exercise 1.
3. Calculate $ÿ\left(1\right)$ where $y\left(t\right)$ is given by:
1.   $t\left({t}^{2}+1\right)$
2.   $sin\left(-2t\right)$
3. $2{e}^{t}+{e}^{2t}$
4. $\frac{1}{t}$
5. $cos\frac{t}{2}$
4. Calculate $\stackrel{\mathrm{...}}{y}\left(-1\right)$ for the functions given in Exercise 3.
1. $6-4{e}^{2x}$
2. $-9sin3x-cosx$
3. $-\frac{1}{4}{x}^{-3∕2}$
4. ${e}^{x}+{e}^{-x}$
5. $2-\frac{1}{{x}^{2}}$
1. $-8{e}^{2x}$
2. $-27cos3x+sinx$
3. $\frac{3}{8}{x}^{-5∕2}$
4. ${e}^{x}-{e}^{-x}$
5. $\frac{2}{{x}^{3}}$
1. $6$
2. $3.6372$
3. $34.9927$
4. $2$
5. $-0.2194$
1. $6$
2. $-3.3292$
3. $1.8184$
4. $-6$
5. $-0.0599$