If ` xy =log (xy) ,then (dy)/(dx)=`

Options

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

Answer

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

### Step-by-Step Solution:

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

2. **Use Product Rule on the Left Side:**
For the left side \(xy\), we apply the product rule:
\[ \frac{d}{dx}(xy) = y + x\frac{dy}{dx} \]

3. **Differentiate the Right Side:**
For the right side \(\log(xy)\), we use the chain rule:
\[ \frac{d}{dx}(\log(xy)) = \frac{1}{xy} \cdot \frac{d}{dx}(xy) = \frac{1}{xy}(y + x\frac{dy}{dx}) \]

4. **Set the Derivatives Equal:**
Now we equate the derivatives from both sides:
\[ y + x\frac{dy}{dx} = \frac{1}{xy}(y + x\frac{dy}{dx}) \]

5. **Multiply Through by \(xy\):**
To eliminate the fraction, multiply both sides by \(xy\):
\[ xy(y + x\frac{dy}{dx}) = y + x\frac{dy}{dx} \]

6. **Rearranging the Equation:**
Expanding the left side gives:
\[ xy^2 + x^2y\frac{dy}{dx} = y + x\frac{dy}{dx} \]
Rearranging terms, we get:
\[ x^2y\frac{dy}{dx} - x\frac{dy}{dx} = y - xy^2 \]

7. **Factor Out \(\frac{dy}{dx}\):**
Factoring out \(\frac{dy}{dx}\) from the left side:
\[ \frac{dy}{dx}(x^2y - x) = y - xy^2 \]

8. **Solve for \(\frac{dy}{dx}\):**
Finally, we can solve for \(\frac{dy}{dx}\):
\[ \frac{dy}{dx} = \frac{y - xy^2}{x^2y - x} \]

9. **Simplifying the Expression:**
We can simplify this further if needed, but we can also check the options provided.

10. **Final Result:**
After simplification, we find:
\[ \frac{dy}{dx} = -\frac{y}{x} \]