If ` x^(y) =3^(x-y),then (dy)/(dx)=`

Options

A. ` (xlog 3-y)/( xlog (3x))`
B. ` (y-xlog 3)/( xlog (3x))`
C. ` (xlog 3-y)/( xlog (3+x))`
D. ` (y-xlog 3)/( xlog (3+x))`

Answer

To solve the problem \( x^y = 3^{x-y} \) and find \( \frac{dy}{dx} \), we will differentiate both sides of the equation with respect to \( x \). Here’s the step-by-step solution:

### Step 1: Differentiate both sides
We start with the equation:
\[ x^y = 3^{x-y} \]
Now, we will differentiate both sides with respect to \( x \).

### Step 2: Differentiate the left side \( x^y \)
To differentiate \( x^y \), we use logarithmic differentiation. Let \( z = x^y \). Taking the natural logarithm of both sides:
\[ \ln z = y \ln x \]
Differentiating both sides with respect to \( x \):
\[ \frac{1}{z} \frac{dz}{dx} = \frac{dy}{dx} \ln x + y \frac{1}{x} \]
Multiplying through by \( z \) (which is \( x^y \)):
\[ \frac{dz}{dx} = x^y \left( \frac{dy}{dx} \ln x + \frac{y}{x} \right) \]

### Step 3: Differentiate the right side \( 3^{x-y} \)
Now, differentiate \( 3^{x-y} \):
\[ \frac{d}{dx}(3^{x-y}) = 3^{x-y} \ln 3 \left(1 - \frac{dy}{dx}\right) \]

### Step 4: Set the derivatives equal
Now we set the derivatives from both sides equal to each other:
\[ x^y \left( \frac{dy}{dx} \ln x + \frac{y}{x} \right) = 3^{x-y} \ln 3 \left( 1 - \frac{dy}{dx} \right) \]

### Step 5: Expand and rearrange
Expanding both sides gives:
\[ x^y \frac{dy}{dx} \ln x + y x^{y-1} = 3^{x-y} \ln 3 - 3^{x-y} \ln 3 \frac{dy}{dx} \]
Now, we will collect all the \( \frac{dy}{dx} \) terms on one side:
\[ x^y \frac{dy}{dx} \ln x + 3^{x-y} \ln 3 \frac{dy}{dx} = 3^{x-y} \ln 3 - y x^{y-1} \]

### Step 6: Factor out \( \frac{dy}{dx} \)
Factoring out \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} \left( x^y \ln x + 3^{x-y} \ln 3 \right) = 3^{x-y} \ln 3 - y x^{y-1} \]

### Step 7: Solve for \( \frac{dy}{dx} \)
Finally, we solve for \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{3^{x-y} \ln 3 - y x^{y-1}}{x^y \ln x + 3^{x-y} \ln 3} \]

### Final Answer
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = \frac{3^{x-y} \ln 3 - y x^{y-1}}{x^y \ln x + 3^{x-y} \ln 3} \]