# How do you use implicit differentiation to find (dy)/(dx) given 3x^2y+2xy^3=1?

Sep 15, 2016

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{- 2 {y}^{3} - 6 x y}{3 {x}^{2} + 6 x {y}^{2}}$

#### Explanation:

differentiate with respect to x, using the $\textcolor{b l u e}{\text{product rule}}$

$3 {x}^{2} \frac{\mathrm{dy}}{\mathrm{dx}} + 6 x y + 2 x .3 {y}^{2} \frac{\mathrm{dy}}{\mathrm{dx}} + 2 {y}^{3} = 0$

$\Rightarrow 3 {x}^{2} \frac{\mathrm{dy}}{\mathrm{dx}} + 6 x y + 6 x {y}^{2} \frac{\mathrm{dy}}{\mathrm{dx}} + 2 {y}^{3} = 0$

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}} \left(3 {x}^{2} + 6 x {y}^{2}\right) = - 2 {y}^{3} - 6 x y$

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{- 2 {y}^{3} - 6 x y}{3 {x}^{2} + 6 x {y}^{2}}$