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

Feb 15, 2017

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

#### Explanation:

Differentiate both sides of the equation with repect to $x$:

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

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

Solve for $\frac{\mathrm{dy}}{\mathrm{dx}}$:

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

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

Feb 15, 2017

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

#### Explanation:

differentiate each term on both sides $\textcolor{b l u e}{\text{implicitly with respect to x}}$

Use the $\textcolor{b l u e}{\text{product rule}}$ on the term xy

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

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

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