# What is the implicit derivative of 5=xy^3-3xy?

Jan 2, 2016

$f ' \left(x\right) = \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{y \left({y}^{2} - 3\right)}{3 x \left(1 - {y}^{2}\right)}$

#### Explanation:

Given: $5 = x {y}^{3} - 3 x y$

We will find the derivative with respect to $\mathrm{dx}$

$\frac{d}{\mathrm{dx}} \left(5\right) = \frac{d}{\mathrm{dx}} \left(x {y}^{3}\right) - \frac{d}{\mathrm{dx}} \left(3 x y\right) \text{ " " " }$ Product rule

0 = d/dx(x) y^3 + x d/dx(y^3) - d/dx((3xy) - 3x(d/dx y)

$0 = {y}^{3} + 3 x {y}^{2} \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) - 3 y - 3 x \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)$

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

Gather like terms $\frac{\mathrm{dy}}{\mathrm{dx}}$ on one side, then factor, solve for $\frac{\mathrm{dy}}{\mathrm{dx}}$

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

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{{y}^{3} - 3 y}{3 x - 3 x {y}^{2}}$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{y \left({y}^{2} - 3\right)}{3 x \left(1 - {y}^{2}\right)}$