What is the derivative of `(log_(tanx)cotx)(log_(cotx)tanx)^(-1)` at `x=(pi)/(4)`?

Options

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

Explanation

To find the derivative of the function \( f(x) = \left( \log_{\tan x} \cot x \right) \left( \log_{\cot x} \tan x \right)^{-1} \) at \( x = \frac{\pi}{4} \), we can follow these steps:

### Step 1: Simplify the Function
First, we can rewrite the logarithmic expressions using the change of base formula:
\[ \log_{\tan x} \cot x = \frac{\log \cot x}{\log \tan x} \]
\[ \log_{\cot x} \tan x = \frac{\log \tan x}{\log \cot x} \]

Thus, we can express \( f(x) \) as:
\[ f(x) = \frac{\log \cot x}{\log \tan x} \cdot \left( \frac{\log \tan x}{\log \cot x} \right)^{-1} \]

### Step 2: Simplify Further
The term \( \left( \frac{\log \tan x}{\log \cot x} \right)^{-1} \) can be rewritten as:
\[ \left( \frac{\log \tan x}{\log \cot x} \right)^{-1} = \frac{\log \cot x}{\log \tan x} \]

Now substituting this back into \( f(x) \):
\[ f(x) = \frac{\log \cot x}{\log \tan x} \cdot \frac{\log \cot x}{\log \tan x} = \left( \frac{\log \cot x}{\log \tan x} \right)^2 \]

### Step 3: Evaluate at \( x = \frac{\pi}{4} \)
At \( x = \frac{\pi}{4} \):
\[ \tan\left(\frac{\pi}{4}\right) = 1 \quad \text{and} \quad \cot\left(\frac{\pi}{4}\right) = 1 \]
Thus:
\[ \log \tan\left(\frac{\pi}{4}\right) = \log 1 = 0 \quad \text{and} \quad \log \cot\left(\frac{\pi}{4}\right) = \log 1 = 0 \]

This means:
\[ f\left(\frac{\pi}{4}\right) = \left( \frac{\log 1}{\log 1} \right)^2 = \left( \frac{0}{0} \right)^2 \]
This is an indeterminate form, so we need to apply L'Hôpital's Rule or analyze the limit.

### Step 4: Find the Derivative
To find the derivative \( f'(x) \), we can use the quotient rule and chain rule. However, since we found that \( f\left(\frac{\pi}{4}\right) \) leads to an indeterminate form, we can analyze the behavior around \( x = \frac{\pi}{4} \).

Given that both \( \log \tan x \) and \( \log \cot x \) approach \( 0 \) as \( x \) approaches \( \frac{\pi}{4} \), we can conclude that \( f(x) \) is constant around this point.

### Conclusion
Thus, the derivative \( f'(x) \) at \( x = \frac{\pi}{4} \) is:
\[ f'\left(\frac{\pi}{4}\right) = 0 \]