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

Feb 18, 2017

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

#### Explanation:

$\left(x - 1\right) {y}^{2} = x + 1$

${y}^{2} = \frac{x + 1}{x - 1}$

Using implicit differentiation and the quotient rule:

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

$= \frac{- 2}{x - 1} ^ 2$

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

$= - \frac{1}{{\left(x - 1\right)}^{2} \cdot y}$