Desivatives of $$ \frac{\sqrt{ an x}}{} $$

Question

Desivatives of $$ \frac{\sqrt{	an x}}{} $$

UpStudy AI · Accepted Answer

To find the derivative of the function $$ y = \sqrt{\tan x} $$, we can use the chain rule.

First, we can rewrite the function as:

$$
y = (\tan x)^{1/2}
$$

Now, we will apply the chain rule. The derivative of $$ y $$ with respect to $$ x $$ is given by:

$$
\frac{dy}{dx} = \frac{1}{2} (\tan x)^{-1/2} \cdot \frac{d}{dx}(\tan x)
$$

Next, we need to find the derivative of $$ \tan x $$:

$$
\frac{d}{dx}(\tan x) = \sec^2 x
$$

Now, substituting this back into our expression for $$ \frac{dy}{dx} $$:

$$
\frac{dy}{dx} = \frac{1}{2} (\tan x)^{-1/2} \cdot \sec^2 x
$$

We can rewrite this as:

$$
\frac{dy}{dx} = \frac{\sec^2 x}{2 \sqrt{\tan x}}
$$

Thus, the derivative of $$ \sqrt{\tan x} $$ is:

$$
\frac{dy}{dx} = \frac{\sec^2 x}{2 \sqrt{\tan x}}
$$
The derivative of $$ \sqrt{\tan x} $$ is $$ \frac{\sec^2 x}{2 \sqrt{\tan x}} $$.

Desivatives of \( \frac{\sqrt{\tan x}}{} \)