# How do you find dy/dx by implicit differentiation of x^2+y^2=36?

Jan 20, 2017

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

#### Explanation:

$\frac{d}{\mathrm{dx}} \left({x}^{2} + {y}^{2}\right) = \frac{d}{\mathrm{dx}} 36$

$\implies \frac{d}{\mathrm{dx}} {x}^{2} + \frac{d}{\mathrm{dx}} {y}^{2} = 0$

$\implies 2 x + 2 y \left(\frac{d}{\mathrm{dx}} y\right) = 0$
(Applying the chain rule with $y$ treated as a function of $x$)

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

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

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