If ` y= x^y `, then ` x(1-y log x) . (dy)/(dx)=`

Options

A. `x^2 `
B. `y^2 `
C. `xy^2 `
D. None of these

Answer

To solve the problem where \( y = x^y \), we will differentiate both sides with respect to \( x \) and manipulate the resulting equation to find \( x(1 - y \log x) \frac{dy}{dx} \).

### Step-by-Step Solution:

1. **Take the logarithm of both sides**:
\[ \log y = \log(x^y) \]

2. **Apply the logarithmic property**:
\[ \log y = y \log x \]

3. **Differentiate both sides with respect to \( x \)**:
- For the left-hand side, using implicit differentiation:
\[ \frac{1}{y} \frac{dy}{dx} \]
- For the right-hand side, using the product rule:
\[ \frac{d}{dx}(y \log x) = \frac{dy}{dx} \log x + y \cdot \frac{1}{x} \]

4. **Set the derivatives equal to each other**:
\[ \frac{1}{y} \frac{dy}{dx} = \frac{dy}{dx} \log x + \frac{y}{x} \]

5. **Rearrange to isolate \( \frac{dy}{dx} \)**:
\[ \frac{1}{y} \frac{dy}{dx} - \frac{dy}{dx} \log x = \frac{y}{x} \]
\[ \frac{dy}{dx} \left( \frac{1}{y} - \log x \right) = \frac{y}{x} \]

6. **Factor out \( \frac{dy}{dx} \)**:
\[ \frac{dy}{dx} = \frac{y/x}{(1/y) - \log x} \]
\[ = \frac{y}{x} \cdot \frac{y}{1 - y \log x} \]

7. **Multiply both sides by \( x(1 - y \log x) \)**:
\[ x(1 - y \log x) \frac{dy}{dx} = y \]

### Final Result:
Thus, we have:
\[ x(1 - y \log x) \frac{dy}{dx} = y \]