# How do you implicitly differentiate 11=(x)/(1-ye^x)?

Consider $f \left(x , y \left(x\right)\right) = 0$ taking the derivative gives
$\frac{\mathrm{df}}{\mathrm{dx}} = \left({f}_{x}\right) + \left({f}_{y}\right) \frac{\mathrm{dy}}{\mathrm{dx}} = 0$ so $\frac{\mathrm{dy}}{\mathrm{dx}} = - {f}_{x} / {f}_{y}$.
Now $f \left(x , y\right) = \frac{x}{1 - y {e}^{x}} - 11$
${f}_{x} = \frac{d}{\mathrm{dx}} \left(\frac{x}{1 - y {e}^{x}} - 11\right) = \frac{1 - {e}^{x} \left(x - 1\right) y}{{e}^{x} y - 1} ^ 2$
${f}_{y} = \frac{d}{\mathrm{dy}} \left(\frac{x}{1 - y {e}^{x}} - 11\right) = \frac{y \left(x - 1\right) - {e}^{-} x}{{e}^{x} y - 1} ^ 2$
finally $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{y \left(1 - x\right) - {e}^{-} x}{x}$