# How do you find the derivative for f(x)=(18x)/(4+(x^2))?

May 9, 2018

$= \frac{- 18 \left({x}^{2} - 4\right)}{{x}^{2} + 4} ^ 2$

#### Explanation:

We're dealing with a rational function here, so we can use the quotient rule stated below:

$f ' \left(x\right) = \frac{g ' \left(x\right) h \left(x\right) - g \left(x\right) h ' \left(x\right)}{h {\left(x\right)}^{2}}$

If we define $\textcolor{b l u e}{g \left(x\right) = 18 x}$ and $\textcolor{p u r p \le}{h \left(x\right) = {x}^{2} + 4}$, we know

$\textcolor{b l u e}{g ' \left(x\right) = 18}$ and $\textcolor{p u r p \le}{h ' \left(x\right) = 2 x}$

Now we can plug in! We get

$f ' \left(x\right) = \frac{\textcolor{b l u e}{18} \textcolor{p u r p \le}{\left({x}^{2} + 4\right)} - \textcolor{b l u e}{18 x} \textcolor{p u r p \le}{\left(2 x\right)}}{{x}^{2} + 4} ^ 2$

$= \frac{- 36 {x}^{2} + 18 {x}^{2} + 72}{{x}^{2} + 4} ^ 2$

=(-18x^2+72)/((x^2+4)^2

$= \frac{- 18 \left({x}^{2} - 4\right)}{{x}^{2} + 4} ^ 2$

Hope this helps!