If ` (cos x)^(y) = (cos y)^(x) , "then" (dy)/(dx) ` is equal to

Options

A. `(log(cos y) + y(tan x))/(log(cos x) + x tan y)`
B. `(log(cos y) - y(tan x))/(log(cos x) - x( tan y))`
C. `(log(tanx) + y(cos y))/(log(cos x) + x( tan y))`
D. None of these

Explanation

To solve the problem \( (\cos x)^y = (\cos y)^x \) and find \( \frac{dy}{dx} \), we will follow these steps:

### Step 1: Take the logarithm of both sides
We start by taking the natural logarithm of both sides of the equation:
\[ \ln((\cos x)^y) = \ln((\cos y)^x) \]

### Step 2: Apply the logarithmic identity
Using the property of logarithms that states \( \ln(a^b) = b \ln(a) \), we can rewrite the equation:
\[ y \ln(\cos x) = x \ln(\cos y) \]

### Step 3: Differentiate both sides with respect to \( x \)
Now we differentiate both sides of the equation with respect to \( x \):
\[ \frac{d}{dx}(y \ln(\cos x)) = \frac{d}{dx}(x \ln(\cos y)) \]

Using the product rule on both sides:
- Left side:
\[ \frac{dy}{dx} \ln(\cos x) + y \frac{d}{dx}(\ln(\cos x)) \]
Using the chain rule for the derivative of \( \ln(\cos x) \):
\[ \frac{d}{dx}(\ln(\cos x)) = \frac{-\sin x}{\cos x} = -\tan x \]
So, the left side becomes:
\[ \frac{dy}{dx} \ln(\cos x) - y \tan x \]

- Right side:
\[ \ln(\cos y) + x \frac{d}{dx}(\ln(\cos y)) \]
Using the chain rule for the derivative of \( \ln(\cos y) \):
\[ \frac{d}{dx}(\ln(\cos y)) = \frac{-\sin y}{\cos y} \frac{dy}{dx} = -\tan y \frac{dy}{dx} \]
So, the right side becomes:
\[ \ln(\cos y) - x \tan y \frac{dy}{dx} \]

### Step 4: Set the derivatives equal to each other
Now we set the derivatives from both sides equal:
\[ \frac{dy}{dx} \ln(\cos x) - y \tan x = \ln(\cos y) - x \tan y \frac{dy}{dx} \]

### Step 5: Collect all \( \frac{dy}{dx} \) terms
Rearranging gives:
\[ \frac{dy}{dx} \ln(\cos x) + x \tan y \frac{dy}{dx} = \ln(\cos y) + y \tan x \]
Factoring out \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} \left( \ln(\cos x) + x \tan y \right) = \ln(\cos y) + y \tan x \]

### Step 6: Solve for \( \frac{dy}{dx} \)
Now we can solve for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{\ln(\cos y) + y \tan x}{\ln(\cos x) + x \tan y} \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = \frac{\ln(\cos y) + y \tan x}{\ln(\cos x) + x \tan y} \]