How do you use the first and second derivatives to sketch y = x / (x^2 - 9)?

May 11, 2018

Explanation:

The function is

$y = \frac{x}{{x}^{2} - 9}$

The domain is $x \in \left(- \infty , - 3\right) \cup \left(- 3 , 3\right) \cup \left(3 , + \infty\right)$

The first derivative is calculated with the quotient rule

$\left(\frac{u}{v}\right) ' = \frac{u ' v - u v '}{{v}^{2}}$

Here,

$u = x$, $\implies$, $u ' = 1$

$v = {x}^{2} - 9$, $\implies$, $v ' = 2 x$

$y ' = \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1 \cdot \left({x}^{2} - 9\right) - \left(x \cdot 2 x\right)}{{x}^{2} - 9} ^ 2$

$= \frac{{x}^{2} - 9 - 2 {x}^{2}}{{x}^{2} - 9} ^ 2$

$= \frac{- 9 - {x}^{2}}{{x}^{2} - 9} ^ 2$

There are no critical points

$y ' \ne 0$, $\forall x \in {D}_{y}$

Let's calculate the second derivative with the quotient rule

$u = - 9 - {x}^{2}$, $\implies$, $u ' = - 2 x$

$v = {\left({x}^{2} - 9\right)}^{2}$, $\implies$, $v ' = 2 \left({x}^{2} - 9\right) \cdot 2 x$

Therefore,

$y ' ' = \frac{{d}^{2} y}{\mathrm{dx}} ^ 2 = \frac{- 2 x {\left({x}^{2} - 9\right)}^{2} - \left(- 9 - {x}^{2}\right) \left(4 x \left({x}^{2} - 9\right)\right)}{{x}^{2} - 9} ^ 4$

$= \frac{- 2 {x}^{3} + 18 x + 36 x + 4 {x}^{3}}{{x}^{2} - 9} ^ 3$

$= \frac{2 {x}^{3} + 54 x}{{x}^{2} - 9} ^ 3$

$= \frac{2 x \left({x}^{2} + 27\right)}{{x}^{2} - 9} ^ 3$

When $y ' ' = 0$, there is a point of inflection.

$\frac{2 x \left({x}^{2} + 27\right)}{{x}^{2} - 9} ^ 3 = 0$

$\implies$, $x = 0$

Let's make a variation chart

$\textcolor{w h i t e}{a}$$\text{ Interval }$$\textcolor{w h i t e}{a a a}$$\left(- \infty , - 3\right)$$\textcolor{w h i t e}{a a a}$$\left(- 3 , 0\right)$$\textcolor{w h i t e}{a a a}$$\left(0 , 3\right)$$\textcolor{w h i t e}{a a a}$$\left(3 , + \infty\right)$

$\textcolor{w h i t e}{a}$$\text{ Sign y'' }$$\textcolor{w h i t e}{a a a a a a a a}$$-$$\textcolor{w h i t e}{a a a a a a a a a}$$+$$\textcolor{w h i t e}{a a a a a a}$$-$$\textcolor{w h i t e}{a a a a a a}$$+$

$\textcolor{w h i t e}{a}$$\text{ y }$$\textcolor{w h i t e}{a a a a a a a a a a a a a}$$\cap$$\textcolor{w h i t e}{a a a a a a a a a a}$$\cup$$\textcolor{w h i t e}{a a a a a a}$$\cap$$\textcolor{w h i t e}{a a a a a a}$$\cup$

graph{x/(x^2-9) [-10, 10, -5, 5]}