Question

What is the derivative of $\sqrt{\tan x}$?

Answer 1

$\frac{{\sec }^{2}x}{2\sqrt{\tan x}}$
Explanation:
The expression can be written as

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

According to the chain rule,

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

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

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