Step-by-step Solution:

logarithm of both sides: \[ m \log x + n \log y = (m + n) \log (x + y). \] Differentiating both sides with respect to \( x \), we get: \[ \frac{m}{x} + \frac{n}{y} \frac{dy}{dx} = \frac{m+n}{x+y}\left(1 + \frac{dy}{dx}\right). \] Simplifying: \[ \frac{m}{x} - \frac{m+n}{x+y} = \left( \frac{m+n}{x+y} - \frac{n}{y} \right) \frac{dy}{dx}. \] This simplifies further to: \[ \frac{m y - n x}{x(x+y)} = \left( \frac{m y - n x}{y(x+y)} \right) \frac{dy}{dx}. \] Hence: \[ \frac{dy}{dx} = \frac{y}{x}. \]