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
$\phantom{\rule{2em}{0ex}}\text{secondderivative=}\frac{d\phantom{\rule{1em}{0ex}}}{dx}\left(\frac{dy}{dx}\right)$
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 ${y}^{\u2033}$ .
In similar manner, the third derivative is denoted by $\frac{{d}^{3}y}{d{x}^{3}}$ or ${y}^{\u2034}$ and so on. So, referring to Example 6 we could have written
$$\begin{array}{rcll}\text{firstderivative}=\frac{dy}{dx}& =& 4{x}^{3}+12x& \text{}\\ \text{secondderivative}=\frac{{d}^{2}y}{d{x}^{2}}& =& 12{x}^{2}+12& \text{}\\ \text{thirdderivative}=\frac{{d}^{3}y}{d{x}^{3}}& =& 24x& \text{}\end{array}$$
Key Point 7
If $y=y\left(x\right)$ then its first, second and third derivatives are denoted by:
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 $\u1e8f$ , pronounced ‘ $y$ dot’. Similarly, a second derivative with respect to $t$ can be written as $\xff$ , pronounced ‘ $y$ double dot’.
Key Point 8
If $y=y\left(t\right)$ then
Task!
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 $\u1e8f=2{e}^{2t}sint$ , $\xff=4{e}^{2t}cost$ and $\xff\phantom{\rule{0.3em}{0ex}}\stackrel{\u0307}{}=8{\text{e}}^{2t}+sint$
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 ${y}^{\u2033}\left(2\right)$ . The third derivative evaluated at $x=1$ is written as $\frac{{d}^{3}y}{d{x}^{3}}\left(1\right)$ or ${y}^{\u2034}\left(1\right)$ .
Task!
Given $y\left(x\right)=2sinx+3{x}^{2}$ find
 ${y}^{\prime}\left(1\right)$
 ${y}^{\u2033}\left(1\right)$
 ${y}^{\u2034}\left(0\right)$
First find ${y}^{\prime}\left(x\right),{y}^{\u2033}\left(x\right)$ and ${y}^{\u2034}\left(x\right)$ :
${y}^{\prime}\left(x\right)=2cosx+6x\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}{y}^{\u2033}\left(x\right)=2sinx+6\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}{y}^{\u2034}\left(x\right)=2cosx$
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 ${y}^{\u2033}\left(1\right)$ :
${y}^{\u2033}\left(1\right)=2sin\left(1\right)+6=7.6829$ Finally, find ${y}^{\u2034}\left(0\right)$ :
${y}^{\u2034}\left(0\right)=2cos0=2$ .
Exercises

Find
$\frac{{d}^{2}y}{d{x}^{2}}$
where
$y\left(x\right)$
is defined by:
 $3{x}^{2}{e}^{2x}$
 $sin3x+cosx$
 $\sqrt{x}$
 ${e}^{x}+{e}^{x}$
 $1+x+{x}^{2}+lnx$
 Find $\frac{{d}^{3}y}{d{x}^{3}}$ where $y$ is given in Exercise 1.

Calculate
$\xff\left(1\right)$
where
$y\left(t\right)$
is given by:
 $t\left({t}^{2}+1\right)$
 $sin\left(2t\right)$
 $2{e}^{t}+{e}^{2t}$
 $\frac{1}{t}$
 $cos\frac{t}{2}$
 Calculate $\stackrel{\mathrm{...}}{y}\left(1\right)$ for the functions given in Exercise 3.

 $64{e}^{2x}$
 $9sin3xcosx$
 $\frac{1}{4}{x}^{3\u22152}$
 ${e}^{x}+{e}^{x}$
 $2\frac{1}{{x}^{2}}$

 $8{e}^{2x}$
 $27cos3x+sinx$
 $\frac{3}{8}{x}^{5\u22152}$
 ${e}^{x}{e}^{x}$
 $\frac{2}{{x}^{3}}$

 $6$
 $3.6372$
 $34.9927$
 $2$
 $0.2194$

 $6$
 $3.3292$
 $1.8184$
 $6$
 $0.0599$