If `y=logsqrt(tanx)" then "(dy)/(dx)" at " x=(pi)/(4)` is

Options

A. `oo`
B. 1
C. 0
D. `(1)/(2)`

Answer

To find \(\frac{dy}{dx}\) 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
Now we will 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 \(\frac{d}{dx}(\tan x) = \sec^2 x\), so substituting this in gives:
\[ \frac{dy}{dx} = \frac{1}{2} \cdot \frac{1}{\tan x} \cdot \sec^2 x \]

### Step 3: Simplify the expression
Using the identity \(\sec^2 x = 1 + \tan^2 x\), we can rewrite \(\frac{dy}{dx}\) as:
\[ \frac{dy}{dx} = \frac{\sec^2 x}{2 \tan x} \]

### Step 4: Evaluate at \(x = \frac{\pi}{4}\)
Now we will evaluate \(\frac{dy}{dx}\) at \(x = \frac{\pi}{4}\):
\[ \tan\left(\frac{\pi}{4}\right) = 1 \quad \text{and} \quad \sec\left(\frac{\pi}{4}\right) = \sqrt{2} \]
Thus,
\[ \frac{dy}{dx} \bigg|_{x = \frac{\pi}{4}} = \frac{\sec^2\left(\frac{\pi}{4}\right)}{2 \tan\left(\frac{\pi}{4}\right)} = \frac{(\sqrt{2})^2}{2 \cdot 1} = \frac{2}{2} = 1 \]

### Final Answer
Therefore, \(\frac{dy}{dx}\) at \(x = \frac{\pi}{4}\) is:
\[ \boxed{1} \]

---