`y= log sqrt(tan x)`, find `(dy)/(dx)` at `x= (pi)/(4)`

Answer

To solve the problem of finding \(\frac{dy}{dx}\) for the function \(y = \log(\sqrt{\tan x})\) at \(x = \frac{\pi}{4}\), we will follow these steps:

### Step 1: Rewrite the function
We start with the function:
\[ y = \log(\sqrt{\tan x}) \]
Using the property of logarithms, we can rewrite this as:
\[ y = \frac{1}{2} \log(\tan x) \]

### Step 2: Differentiate using the chain rule
To find \(\frac{dy}{dx}\), we differentiate \(y\) with respect to \(x\):
\[ \frac{dy}{dx} = \frac{1}{2} \cdot \frac{1}{\tan x} \cdot \frac{d}{dx}(\tan x) \]
We know that the derivative of \(\tan x\) is \(\sec^2 x\). Therefore, we have:
\[ \frac{dy}{dx} = \frac{1}{2} \cdot \frac{1}{\tan x} \cdot \sec^2 x \]

### Step 3: Simplify the expression
We can simplify the expression:
\[ \frac{dy}{dx} = \frac{\sec^2 x}{2 \tan x} \]

### Step 4: Evaluate at \(x = \frac{\pi}{4}\)
Now we need to evaluate \(\frac{dy}{dx}\) at \(x = \frac{\pi}{4}\):
- We know that \(\tan\left(\frac{\pi}{4}\right) = 1\)
- We also know that \(\sec\left(\frac{\pi}{4}\right) = \frac{1}{\cos\left(\frac{\pi}{4}\right)} = \frac{1}{\frac{1}{\sqrt{2}}} = \sqrt{2}\), so \(\sec^2\left(\frac{\pi}{4}\right) = 2\)

Substituting these values into our derivative:
\[ \frac{dy}{dx} \bigg|_{x = \frac{\pi}{4}} = \frac{2}{2 \cdot 1} = \frac{2}{2} = 1 \]

### Final Answer
Thus, the value of \(\frac{dy}{dx}\) at \(x = \frac{\pi}{4}\) is:
\[ \frac{dy}{dx} = 1 \]