# How do you differentiate x / (x^2 + 1)^(1/2)?

May 13, 2018

$\frac{d}{\mathrm{dx}} \left(\frac{x}{{x}^{2} + 1} ^ \left(\frac{1}{2}\right)\right) = \frac{1}{{x}^{2} + 1} ^ \left(\frac{3}{2}\right)$

#### Explanation:

Using the quotient rule:

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

$\frac{d}{\mathrm{dx}} \left(\frac{x}{{x}^{2} + 1} ^ \left(\frac{1}{2}\right)\right) = \frac{{x}^{2} + 1 - {x}^{2}}{{x}^{2} + 1} ^ \left(\frac{3}{2}\right)$

$\frac{d}{\mathrm{dx}} \left(\frac{x}{{x}^{2} + 1} ^ \left(\frac{1}{2}\right)\right) = \frac{1}{{x}^{2} + 1} ^ \left(\frac{3}{2}\right)$