# How do you use implicit differentiation to find dy/dx given y^2+xlny=3?

##### 1 Answer
Sep 11, 2016

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{y \ln y}{2 {y}^{2} + x}$

#### Explanation:

$\frac{d}{\mathrm{dx}} \left({y}^{2} + x \ln y\right) = \frac{d}{\mathrm{dx}} \left(3\right)$

$\frac{d}{\mathrm{dx}} \left({y}^{2}\right) + \frac{d}{\mathrm{dx}} \left(x \ln y\right) = \frac{d}{\mathrm{dx}} \left(3\right)$

$2 y \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) + 1 \ln y + x \left(\frac{1}{y}\right) \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) = 0$

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

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

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \ln \frac{y}{2 y + \frac{x}{y}}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{y \ln y}{2 {y}^{2} + x}$

Hopefully this helps!