What is the implicit derivative of -2=xy^2-3xy?

Nov 30, 2015

I found: $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{y \left(3 - y\right)}{x \left(2 y - 3\right)}$

Explanation:

The idea is that you need to derive $y$ as well as it represents a function of $x$; so, for example to derive ${y}^{2}$ you get: $2 y \frac{\mathrm{dy}}{\mathrm{dx}}$!
Where the $\frac{\mathrm{dy}}{\mathrm{dx}}$ bit takes into accont the $x$ dependence of $y$:

$0 = 1 \cdot {y}^{2} + 2 x y \frac{\mathrm{dy}}{\mathrm{dx}} - 3 y - 3 x \cdot 1 \frac{\mathrm{dy}}{\mathrm{dx}}$
collect: $\frac{\mathrm{dy}}{\mathrm{dx}}$:
$\frac{\mathrm{dy}}{\mathrm{dx}} \left[2 x y - 3 x\right] = 3 y - {y}^{2}$
$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{y \left(3 - y\right)}{x \left(2 y - 3\right)}$