If xe^y = 1, then the value of dy/dx at x = 1 is:

The correct answer is Option (1) → $-1$ ##

We have $xe^y = 1$.

Differentiating w.r.t. '$x$' both sides:

$1 \cdot e^y + xe^y \frac{dy}{dx} = 0$

$xe^y \frac{dy}{dx} = -e^y$

$\frac{dy}{dx} = -\frac{1}{x}$

$∴\left. \frac{dy}{dx} \right|_{x=1} = -1$