`y=(tan x ^ tanx)^(tanx)`, then `(dy)/(dx)` =1 at `x = pi/4`

Answer

To find the derivative of the function \( y = (\tan x)^{\tan x} \) and check if \( \frac{dy}{dx} = 1 \) at \( x = \frac{\pi}{4} \), we will follow these steps:

### Step 1: Rewrite the function using logarithms
We start by taking the natural logarithm of both sides to simplify the differentiation process.

\[ \ln y = \ln \left( (\tan x)^{\tan x} \right) \]

Using the property of logarithms, we can rewrite this as:

\[ \ln y = \tan x \cdot \ln(\tan x) \]

### Step 2: Differentiate both sides
Now we differentiate both sides with respect to \( x \). We will use implicit differentiation on the left side and the product rule on the right side.

\[ \frac{d}{dx}(\ln y) = \frac{1}{y} \frac{dy}{dx} \]

For the right side, we apply the product rule:

\[ \frac{d}{dx}(\tan x \cdot \ln(\tan x)) = \frac{d}{dx}(\tan x) \cdot \ln(\tan x) + \tan x \cdot \frac{d}{dx}(\ln(\tan x)) \]

Calculating the derivatives:

1. \( \frac{d}{dx}(\tan x) = \sec^2 x \)
2. \( \frac{d}{dx}(\ln(\tan x)) = \frac{1}{\tan x} \cdot \sec^2 x = \frac{\sec^2 x}{\tan x} \)

So, we have:

\[ \frac{dy}{dx} \cdot \frac{1}{y} = \sec^2 x \cdot \ln(\tan x) + \tan x \cdot \frac{\sec^2 x}{\tan x} \]

This simplifies to:

\[ \frac{dy}{dx} \cdot \frac{1}{y} = \sec^2 x \cdot \ln(\tan x) + \sec^2 x \]

### Step 3: Solve for \( \frac{dy}{dx} \)
Now we can isolate \( \frac{dy}{dx} \):

\[ \frac{dy}{dx} = y \left( \sec^2 x \cdot \ln(\tan x) + \sec^2 x \right) \]

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 4: Evaluate at \( x = \frac{\pi}{4} \)
Now we need to evaluate \( \frac{dy}{dx} \) at \( x = \frac{\pi}{4} \):

1. \( \tan\left(\frac{\pi}{4}\right) = 1 \)
2. \( \sec^2\left(\frac{\pi}{4}\right) = 2 \)
3. \( \ln(\tan\left(\frac{\pi}{4}\right)) = \ln(1) = 0 \)

Substituting these values into the derivative:

\[ \frac{dy}{dx} = (1)^{1} \left( 2 \cdot 0 + 2 \right) = 1 \cdot 2 = 2 \]

### Conclusion
Thus, \( \frac{dy}{dx} \) at \( x = \frac{\pi}{4} \) is 2, not 1. Therefore, the statement is **false**.

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