# How do you find the derivative of ((2x+1)^(5)/(x^(2)+1)^(1/2))?

##### 1 Answer
May 7, 2017

$\therefore f ' \left(x\right) = \left(\frac{{\left(2 x + 1\right)}^{5}}{{\left({x}^{2} + 1\right)}^{\frac{1}{2}}}\right) \left[\frac{10}{2 x + 1} - \frac{x}{{x}^{2} + 1}\right]$

#### Explanation:

Let $f \left(x\right) = \left(\frac{{\left(2 x + 1\right)}^{5}}{{\left({x}^{2} + 1\right)}^{\frac{1}{2}}}\right)$
$\therefore \ln f \left(x\right) = 5 \ln \left(2 x + 1\right) - \frac{1}{2} \ln \left({x}^{2} + 1\right)$
Now differentiate with respect to $x$
$\frac{f ' \left(x\right)}{f \left(x\right)} = \frac{10}{2 x + 1} - \frac{x}{{x}^{2} + 1}$
$\therefore f ' \left(x\right) = \left(\frac{{\left(2 x + 1\right)}^{5}}{{\left({x}^{2} + 1\right)}^{\frac{1}{2}}}\right) \left[\frac{10}{2 x + 1} - \frac{x}{{x}^{2} + 1}\right]$