If ` x^(y) y^(x)=(x+ y)^(x+y).then (dy)/(dx)=`

Options

A. ` -((y+xlog y -x(1+log (x+ y)))/( x+y log x -y (1+log (x+y ))))`
B. ` (y+xlog y -x(1+log (x+ y)))/( x+y log x -y (1+log (x+y )))`
C. `(y/x) (-y-xlog y +x(1+log (x+ y)))/( x+y log x -y (1+log (x+y )))`
D. ` (y/x)(y+xlog y -x(1+log (x+ y)))/( x+y log x -y (1+log (x+y )))`

Answer

To solve the equation \( x^y y^x = (x+y)^{x+y} \) and find \( \frac{dy}{dx} \), we will use logarithmic differentiation. Here’s a step-by-step solution:

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

### Step 2: Apply logarithmic properties
Using the properties of logarithms, we can simplify both sides:
\[ \log(x^y) + \log(y^x) = (x+y) \log(x+y) \]
This can be further simplified to:
\[ y \log x + x \log y = (x+y) \log(x+y) \]

### Step 3: Differentiate both sides with respect to \( x \)
Now we differentiate both sides with respect to \( x \):
\[ \frac{d}{dx}(y \log x) + \frac{d}{dx}(x \log y) = \frac{d}{dx}((x+y) \log(x+y)) \]

Using the product rule on the left side:
\[ \frac{dy}{dx} \log x + y \cdot \frac{1}{x} + \log y + x \cdot \frac{dy}{dx} \cdot \frac{1}{y} = \frac{d}{dx}((x+y) \log(x+y)) \]

### Step 4: Differentiate the right side
For the right side, we apply the product rule:
\[ \frac{d}{dx}((x+y) \log(x+y)) = \log(x+y) + (x+y) \cdot \frac{1}{x+y} \cdot (1 + \frac{dy}{dx}) \]
This simplifies to:
\[ \log(x+y) + (1 + \frac{dy}{dx}) \]

### Step 5: Set up the equation
Now we have:
\[ \frac{dy}{dx} \log x + y \cdot \frac{1}{x} + \log y + x \cdot \frac{dy}{dx} \cdot \frac{1}{y} = \log(x+y) + 1 + \frac{dy}{dx} \]

### Step 6: Collect terms involving \( \frac{dy}{dx} \)
Rearranging gives:
\[ \frac{dy}{dx} \log x + \frac{dy}{dx} \cdot \frac{x}{y} - \frac{dy}{dx} = \log(x+y) + 1 - y \cdot \frac{1}{x} - \log y \]

### Step 7: Factor out \( \frac{dy}{dx} \)
Factoring out \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} \left( \log x + \frac{x}{y} - 1 \right) = \log(x+y) + 1 - \frac{y}{x} - \log y \]

### Step 8: Solve for \( \frac{dy}{dx} \)
Now, isolate \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{\log(x+y) + 1 - \frac{y}{x} - \log y}{\log x + \frac{x}{y} - 1} \]

### Final Result
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = \frac{\log(x+y) + 1 - \frac{y}{x} - \log y}{\log x + \frac{x}{y} - 1} \]