# How do you differentiate y=x^2y-y^2x?

Jul 6, 2016

I found: $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{2 x y - {y}^{2}}{- {x}^{2} + 2 y x + 1}$

#### Explanation:

We use Implicit Differentiation where we remember that $y$ is a function of $x$ and must be derived accordingly;
for example if you have ${y}^{2}$ you derive it getting: $2 y \frac{\mathrm{dy}}{\mathrm{dx}}$ to take into account this depndence.

$1 \frac{\mathrm{dy}}{\mathrm{dx}} = 2 x y + {x}^{2} \frac{\mathrm{dy}}{\mathrm{dx}} - 2 y x \frac{\mathrm{dy}}{\mathrm{dx}} - {y}^{2}$
collect $\frac{\mathrm{dy}}{\mathrm{dx}}$ and rearrange:
$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{2 x y - {y}^{2}}{- {x}^{2} + 2 y x + 1}$