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

Oct 24, 2017

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

#### Explanation:

we want

$\frac{d}{\mathrm{dx}} \left({x}^{2} + {y}^{2} = 2\right)$

we differentiate each term $w r t \text{ } x$and when we differentiate $y$ we just multiply that term by $\frac{\mathrm{dy}}{\mathrm{dx}}$ by virtue of the chain rule

$\frac{d}{\mathrm{dx}} \left({x}^{2} + {y}^{2} = 2\right)$

$\implies \frac{d}{\mathrm{dx}} \left({x}^{2}\right) + \frac{d}{\mathrm{dx}} \left({y}^{2}\right) = \frac{d}{\mathrm{dx}} \left(2\right)$

$2 x + 2 y \frac{\mathrm{dy}}{\mathrm{dx}} = 0$

now rearrange for $\frac{\mathrm{dy}}{\mathrm{dx}}$

2y(dy)/(dx)=-2x#

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