Answer
$\frac{dy}{dx} = \frac{x^{2}(1 - y^{2}) + ye^{\frac{y}{x}}}{x(e^{\frac{x}{y}} + 2x^{2}y)}$
Work Step by Step
$e^{\frac{x}{y}} + xy^{2} - x = c$
Implicit differentiation: differentiating the equation in terms of $x$:
$e^{\frac{y}{x}}\frac{x\frac{dy}{dx} - y}{x^{2}} + 2xy\frac{dy}{dx} + y^{2} - 1 = 0$
Express the equation in terms of $\frac{dy}{dx}$;
$\frac{dy}{dx} = \frac{x^{2}(1 - y^{2}) + ye^{\frac{y}{x}}}{x(e^{\frac{x}{y}} + 2x^{2}y)}$
