# What is the derivative of (-x^2+5)/(x^2+5)^2?

Jul 20, 2017

d/(dx) [(-x^2 + 5)/((x^2 + 5)^2)] = color(blue)((2x(x^2-15))/((x^2+5)^3)

#### Explanation:

We're asked to find the derivative

$\frac{d}{\mathrm{dx}} \left[\frac{- {x}^{2} + 5}{{\left({x}^{2} + 5\right)}^{2}}\right]$

Use the quotient rule, which states

$\frac{d}{\mathrm{dx}} \left[\frac{u}{v}\right] = \frac{v \frac{\mathrm{du}}{\mathrm{dx}} - u \frac{\mathrm{dv}}{\mathrm{dx}}}{{v}^{2}}$

where

$u = - {x}^{2} + 5$

$v = {\left({x}^{2} + 5\right)}^{2}$:

$= \frac{{\left({x}^{2} + 5\right)}^{2} \frac{d}{\mathrm{dx}} \left[- {x}^{2} + 5\right] - \left(- {x}^{2} + 5\right) \frac{d}{\mathrm{dx}} \left[{\left({x}^{2} + 5\right)}^{2}\right]}{{\left({\left({x}^{2} + 5\right)}^{2}\right)}^{2}}$

Using power rule on first term:

$= \frac{{\left({x}^{2} + 5\right)}^{2} \left(- 2 x\right) - \left(- {x}^{2} + 5\right) \frac{d}{\mathrm{dx}} \left[{\left({x}^{2} + 5\right)}^{2}\right]}{{\left({x}^{2} + 5\right)}^{4}}$

Use the chain rule to differentiate the second term, which in this case is

$\frac{d}{\mathrm{dx}} \left[{\left({x}^{2} + 5\right)}^{2}\right] = \frac{d}{\mathrm{du}} \left[{u}^{2}\right] \frac{\mathrm{du}}{\mathrm{dx}}$

where

$u = {x}^{2} + 5$

$\frac{d}{\mathrm{du}} \left[{u}^{2}\right] = 2 u$ (from power rule):

$= \frac{{\left({x}^{2} + 5\right)}^{2} \left(- 2 x\right) - \left(- {x}^{2} + 5\right) \left(2 \left({x}^{2} + 5\right)\right) \frac{d}{\mathrm{dx}} \left[{x}^{2} + 5\right]}{{\left({x}^{2} + 5\right)}^{4}}$

$= \frac{{\left({x}^{2} + 5\right)}^{2} \left(- 2 x\right) - \left(- {x}^{2} + 5\right) \left(2 {x}^{2} + 10\right) \left(2 x\right)}{{\left({x}^{2} + 5\right)}^{4}}$

We can simplify this further if we wanted to:

$= \frac{{\left({x}^{2} + 5\right)}^{2} \left(- 2 x\right) - \left(- 2\right) \left({x}^{2} - 5\right) \left({x}^{2} + 5\right) \left(2 x\right)}{{\left({x}^{2} + 5\right)}^{4}}$

Divide both sides by ${x}^{2} + 5$:

$= \frac{\left({x}^{2} + 5\right) \left(- 2 x\right) - \left(- 2\right) \left({x}^{2} - 5\right) \left(2 x\right)}{{\left({x}^{2} + 5\right)}^{3}}$

$= \frac{- 2 {x}^{3} - 10 x - \left(- 2\right) \left({x}^{2} - 5\right) \left(2 x\right)}{{\left({x}^{2} + 5\right)}^{3}}$

$= \frac{- 2 {x}^{3} - 10 x - \left(20 x - 4 {x}^{3}\right)}{{\left({x}^{2} + 5\right)}^{3}}$

$= \frac{2 {x}^{3} - 30 x}{{\left({x}^{2} + 5\right)}^{3}}$

= color(blue)((2x(x^2-15))/((x^2+5)^3)