If `y=[(tanx)^(tanx)]^(tanx)`,then at `x=pi/4` , the value of `(dy)/(dx)=`

Options

A. 0
B. 1
C. 2
D. None of these

Answer

To find the derivative \(\frac{dy}{dx}\) for the function \(y = (\tan x)^{\tan x}\) at \(x = \frac{\pi}{4}\), we will follow these steps:

### Step 1: Rewrite the function
Given:
\[ y = (\tan x)^{\tan x} \]
We can express this as:
\[ y = e^{\tan x \cdot \ln(\tan x)} \]

### Step 2: Take the natural logarithm
Taking the natural logarithm on both sides:
\[ \ln y = \tan x \cdot \ln(\tan x) \]

### Step 3: Differentiate both sides
Now, we differentiate both sides with respect to \(x\):
Using the product rule on the right side:
\[ \frac{1}{y} \frac{dy}{dx} = \sec^2 x \cdot \ln(\tan x) + \tan x \cdot \frac{1}{\tan x} \cdot \sec^2 x \]
This simplifies to:
\[ \frac{1}{y} \frac{dy}{dx} = \sec^2 x \cdot \ln(\tan x) + \sec^2 x \]

### Step 4: Solve for \(\frac{dy}{dx}\)
Multiplying both sides by \(y\):
\[ \frac{dy}{dx} = y \left(\sec^2 x \cdot \ln(\tan x) + \sec^2 x\right) \]
Substituting back \(y = (\tan x)^{\tan x}\):
\[ \frac{dy}{dx} = (\tan x)^{\tan x} \left(\sec^2 x \cdot \ln(\tan x) + \sec^2 x\right) \]

### Step 5: Evaluate at \(x = \frac{\pi}{4}\)
Now we evaluate \(\frac{dy}{dx}\) at \(x = \frac{\pi}{4}\):
- \(\tan\left(\frac{\pi}{4}\right) = 1\)
- \(\sec\left(\frac{\pi}{4}\right) = \sqrt{2}\)
- \(\ln(1) = 0\)

Substituting these values:
\[ \frac{dy}{dx} = (1)^{1} \left(\sqrt{2} \cdot 0 + \sqrt{2}\right) \]
This simplifies to:
\[ \frac{dy}{dx} = \sqrt{2} \]

### Final Answer
Thus, the value of \(\frac{dy}{dx}\) at \(x = \frac{\pi}{4}\) is:
\[ \frac{dy}{dx} = \sqrt{2} \]