If `x^(m) y^(n) =2(x+y)^(m+n)`, the value of `(dy)/(dx)` is

Options

A. x+y
B. `x/y`
C. `y/x`
D. `x-y`

Answer

To solve the equation \( x^m y^n = 2(x+y)^{m+n} \) and find the value of \( \frac{dy}{dx} \), we will follow these steps:

### Step-by-Step Solution:

1. **Take the logarithm of both sides**:
\[ \log(x^m y^n) = \log(2(x+y)^{m+n}) \]

2. **Apply the properties of logarithms**:
Using the property \( \log(a \cdot b) = \log a + \log b \) and \( \log(a^b) = b \log a \):
\[ m \log x + n \log y = \log 2 + (m+n) \log(x+y) \]

3. **Differentiate both sides with respect to \( x \)**:
\[ \frac{d}{dx}(m \log x + n \log y) = \frac{d}{dx}(\log 2 + (m+n) \log(x+y)) \]

The left side becomes:
\[ \frac{m}{x} + n \frac{dy}{dx} \cdot \frac{1}{y} \]

The right side becomes:
\[ 0 + (m+n) \cdot \frac{1}{x+y} \cdot \left(1 + \frac{dy}{dx}\right) \]

4. **Set the derivatives equal**:
\[ \frac{m}{x} + n \frac{dy}{dx} \cdot \frac{1}{y} = \frac{(m+n)}{x+y} \cdot (1 + \frac{dy}{dx}) \]

5. **Multiply through by \( xy(x+y) \) to eliminate denominators**:
\[ m y (x+y) + n x \frac{dy}{dx} (x+y) = (m+n) y (1 + \frac{dy}{dx}) x \]

6. **Rearranging the equation**:
\[ m y (x+y) = (m+n) y x + n x (x+y) \frac{dy}{dx} - (m+n) y x \frac{dy}{dx} \]

Combine terms involving \( \frac{dy}{dx} \):
\[ m y (x+y) - (m+n) y x = (n x (x+y) - (m+n) y x) \frac{dy}{dx} \]

7. **Solve for \( \frac{dy}{dx} \)**:
\[ \frac{dy}{dx} = \frac{m y (x+y) - (m+n) y x}{n x (x+y) - (m+n) y x} \]

8. **Simplifying the expression**:
\[ \frac{dy}{dx} = \frac{m y (x+y) - (m+n) y x}{(n x - (m+n) y) x + n y x} \]

### Final Result:
Thus, the value of \( \frac{dy}{dx} \) is:
\[ \frac{dy}{dx} = \frac{m y (x+y) - (m+n) y x}{n x (x+y) - (m+n) y x} \]