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

$\text{second derivative = }\frac{d}{dx}\left(\frac{dy}{dx}\right)$

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 $y^{″}$ .

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

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’.

Task!

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

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

Answer

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Finally, obtain the third derivative:

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

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 $y^{‴}\left(-1\right)$ .

Task!

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

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

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Now find $y^{″}\left(-1\right)$ :

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Exercises

1. Find $\frac{d^{2}y}{d{x}^2}$ where $y\left(x\right)$ is defined by:
   1. $3x^2-e^{2x}$   
   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. $2e^t+e^{2t}$   
   4. $\frac{1}{t}$   
   5. $cos\frac{t}{2}$
4. Calculate $\stackrel{\mathrm{...}}{y}\left(-1\right)$ for the functions given in Exercise 3.

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