Question

Differentiate
$\sqrt{\tan x}$

• A. $\frac{{\sec }^{2}x}{2\sqrt{\tan x}}$
• B. $\frac{-{\sec }^{2}x}{2\sqrt{\tan x}}$
• C. $\frac{{\csc }^{2}x}{2\sqrt{\tan x}}$
• D. $\frac{-{\csc }^{2}x}{2\sqrt{\tan x}}$

Answer 1

The correct option is A $\frac{{\sec }^{2}x}{2\sqrt{\tan x}}$

$\sqrt{\tan x}=(\tan x{)}^{\frac{1}{2}}$

According to the chain rule, we have

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

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

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