# How do you find the derivative of e^y = xy ^2?

Aug 18, 2016

Differentiate implicitly:

$\frac{d}{\mathrm{dx}} \left({e}^{y}\right) = \frac{d}{\mathrm{dx}} \left(x {y}^{2}\right)$

${e}^{y} \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) = {y}^{2} + 2 x y \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)$

${e}^{y} \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) - 2 x y \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) = {y}^{2}$

$\frac{\mathrm{dy}}{\mathrm{dx}} \left({e}^{y} - 2 x y\right) = {y}^{2}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = {y}^{2} / \left({e}^{y} - 2 x y\right)$

Hopefully this helps!