xy = log (xy)

Options

A. `(dy)/(dx)=-(y)/(x)`
B. `(dy)/(dx)=(y)/(x)`
C. `(dy)/(dx)=-(x)/(y)`
D. `(dy)/(dx)=(x)/(y)`

Explanation

To solve the equation \( xy = \log(xy) \) and find the derivative \( \frac{dy}{dx} \), we will differentiate both sides with respect to \( x \).

### Step-by-Step Solution:

1. **Differentiate both sides**:
We start with the equation:
\[ xy = \log(xy) \]
We will differentiate both sides with respect to \( x \).

2. **Differentiate the left-hand side**:
Using the product rule on the left side:
\[ \frac{d}{dx}(xy) = x \frac{dy}{dx} + y \]

3. **Differentiate the right-hand side**:
For the right side, we use the chain rule:
\[ \frac{d}{dx}(\log(xy)) = \frac{1}{xy} \cdot \frac{d}{dx}(xy) \]
From the previous step, we know that \( \frac{d}{dx}(xy) = x \frac{dy}{dx} + y \). Thus:
\[ \frac{d}{dx}(\log(xy)) = \frac{1}{xy}(x \frac{dy}{dx} + y) \]

4. **Set the derivatives equal**:
Now we set the derivatives from both sides equal to each other:
\[ x \frac{dy}{dx} + y = \frac{1}{xy}(x \frac{dy}{dx} + y) \]

5. **Multiply through by \( xy \) to eliminate the fraction**:
\[ xy(x \frac{dy}{dx} + y) = x \frac{dy}{dx} + y \]
This simplifies to:
\[ x^2y \frac{dy}{dx} + y^2x = x \frac{dy}{dx} + y \]

6. **Rearrange the equation**:
Move all terms involving \( \frac{dy}{dx} \) to one side:
\[ x^2y \frac{dy}{dx} - x \frac{dy}{dx} = y - y^2x \]
Factor out \( \frac{dy}{dx} \):
\[ \left(x^2y - x\right) \frac{dy}{dx} = y(1 - yx) \]

7. **Solve for \( \frac{dy}{dx} \)**:
Finally, we can isolate \( \frac{dy}{dx} \):
\[ \frac{dy}{dx} = \frac{y(1 - yx)}{x^2y - x} \]

### Final Result:
Thus, the derivative \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = \frac{y(1 - yx)}{x^2y - x} \]