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

Apr 12, 2018

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

#### Explanation:

Implicit differentiation of the circle $3 {x}^{2} + {3}^{y} ^ 2 = 2$ gives

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

Solving for $\frac{\mathrm{dy}}{\mathrm{dx}}$ gives

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

Apr 12, 2018

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

#### Explanation:

$\textcolor{b l u e}{\text{differentiate implicitly with respect to x}}$

$\text{noting that}$

$\frac{d}{\mathrm{dx}} \left(y\right) = \frac{\mathrm{dy}}{\mathrm{dx}} \text{ and } \frac{d}{\mathrm{dx}} \left({y}^{2}\right) = 2 y \frac{\mathrm{dy}}{\mathrm{dx}}$

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

$\Rightarrow 6 y \frac{\mathrm{dy}}{\mathrm{dx}} = - 6 x$

$\Rightarrow \frac{\mathrm{dy}}{\mathrm{dx}} = \frac{- 6 x}{6 y} = - \frac{x}{y}$