# What is the derivative of sqrt(tan x)?

Dec 19, 2015

${\sec}^{2} \frac{x}{2 \sqrt{\tan} x}$

#### Explanation:

The expression can be written as

${\left(\tan x\right)}^{\frac{1}{2}}$

According to the chain rule,

$\frac{d}{\mathrm{dx}} \left[{\left(\tan x\right)}^{\frac{1}{2}}\right] = \frac{1}{2} {\left(\tan x\right)}^{- \frac{1}{2}} \cdot \frac{d}{\mathrm{dx}} \left[\tan x\right]$

Since $\frac{d}{\mathrm{dx}} \left[\tan x\right] = {\sec}^{2} x$,

$\frac{d}{\mathrm{dx}} \left[{\left(\tan x\right)}^{\frac{1}{2}}\right] = {\sec}^{2} \frac{x}{2 \sqrt{\tan} x}$