# How do you find the derivative of x/(x^2-4)?

Aug 11, 2015

The answer is in the explanation section.

#### Explanation:

$y = \frac{x}{{x}^{2} - 4}$

$\frac{\mathrm{dy}}{\mathrm{dx}}$ = $\frac{\left(\left({x}^{2} - 4\right) \left(1\right)\right) - \left(\left(x\right) \left(2 x\right)\right)}{{x}^{2} - 4} ^ 2$ [by Quotient Rule]

$\frac{\mathrm{dy}}{\mathrm{dx}}$ = $\frac{\left({x}^{2} - 4\right) - \left(2 {x}^{2}\right)}{{x}^{2} - 4} ^ 2$

$\frac{\mathrm{dy}}{\mathrm{dx}}$ = $\frac{{x}^{2} - 4 - 2 {x}^{2}}{{x}^{2} - 4} ^ 2$

$\frac{\mathrm{dy}}{\mathrm{dx}}$ = $\frac{- {x}^{2} - 4}{{x}^{2} - 4} ^ 2$

Aug 11, 2015

Use the product, power and chain rules to find:

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

#### Explanation:

$\frac{d}{\mathrm{dx}} \left(\frac{x}{{x}^{2} - 4}\right) = \frac{d}{\mathrm{dx}} x {\left({x}^{2} - 4\right)}^{-} 1$

$= \left(\frac{d}{\mathrm{dx}} x\right) {\left({x}^{2} - 4\right)}^{-} 1 + x \left(\frac{d}{\mathrm{dx}} {\left({x}^{2} - 4\right)}^{-} 1\right)$ [Product Rule]

$= {\left({x}^{2} - 4\right)}^{-} 1 - 2 {x}^{2} {\left({x}^{2} - 4\right)}^{-} 2$ [Power Rule and Chain Rule]

$= \frac{\left({x}^{2} - 4\right) - 2 {x}^{2}}{{x}^{2} - 4} ^ 2$

$= - \frac{{x}^{2} + 4}{{x}^{2} - 4} ^ 2$