If `x ylog(x+y)=1`,prove that`(dy)/(dx)=-(y(x^2y+x+y))/(x(x y^2+x+y))dot`

Explanation

To prove that \(\frac{dy}{dx} = -\frac{y(x^2y + x + y)}{x(xy^2 + x + y)}\) given the equation \(xy \log(x+y) = 1\), we will differentiate both sides of the equation with respect to \(x\) using implicit differentiation.

### Step-by-step Solution:

1. **Differentiate the given equation:**
\[ xy \log(x+y) = 1 \]
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