# What is the implicit derivative of 5=-yx^2-xy/(y-1)+y^2x?

Jan 20, 2018

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

#### Explanation:

First we take the change in $x$ of both sides.

$\frac{d}{\mathrm{dx}} \left[5\right] = \frac{d}{\mathrm{dx}} \left[- y {x}^{2}\right] - \frac{d}{\mathrm{dx}} \left[\frac{x y}{y - 1}\right] + \frac{d}{\mathrm{dx}} \left[{y}^{2} x\right]$

$0 = - y \frac{d}{\mathrm{dx}} \left[{x}^{2}\right] + {x}^{2} \frac{d}{\mathrm{dx}} \left[- y\right] - \frac{y}{y - 1} \frac{d}{\mathrm{dx}} \left[x\right] + x \frac{d}{\mathrm{dx}} \left[\frac{y}{y - 1}\right] + {y}^{2} \frac{d}{\mathrm{dx}} \left[x\right] + x \frac{d}{\mathrm{dx}} \left[{y}^{2}\right]$

$0 = - y \left(2 x\right) + {x}^{2} \frac{d}{\mathrm{dx}} \left[- y\right] - \frac{y}{y - 1} \left(1\right) + x \frac{d}{\mathrm{dx}} \left[\frac{y}{y - 1}\right] + {y}^{2} \left(1\right) + x \frac{d}{\mathrm{dx}} \left[{y}^{2}\right]$

$0 = - 2 x y + {x}^{2} \frac{d}{\mathrm{dx}} \left[- y\right] - \frac{y}{y - 1} + x \frac{d}{\mathrm{dx}} \left[\frac{y}{y - 1}\right] + {y}^{2} + x \frac{d}{\mathrm{dx}} \left[{y}^{2}\right]$

The chain rule tells us that $\frac{d}{\mathrm{dx}} = \frac{d}{\mathrm{dy}} \times \frac{\mathrm{dy}}{\mathrm{dx}}$. That gives us:
$0 = - 2 x y + {x}^{2} \frac{\mathrm{dy}}{\mathrm{dx}} \frac{d}{\mathrm{dy}} \left[- y\right] - \frac{y}{y - 1} + x \frac{\mathrm{dy}}{\mathrm{dx}} \frac{d}{\mathrm{dy}} \left[\frac{y}{y - 1}\right] + {y}^{2} + x \frac{\mathrm{dy}}{\mathrm{dx}} \frac{d}{\mathrm{dy}} \left[{y}^{2}\right]$

$0 = - 2 x y + \frac{\mathrm{dy}}{\mathrm{dx}} \left(- {x}^{2}\right) - \frac{y}{y - 1} + \frac{\mathrm{dy}}{\mathrm{dx}} x \left(\frac{\left(y - 1\right) \frac{d}{\mathrm{dy}} \left(y\right) - \left(y \frac{d}{\mathrm{dy}} \left(y - 1\right)\right)}{y - 1} ^ 2\right) + {y}^{2} + \frac{\mathrm{dy}}{\mathrm{dx}} \left(2 y x\right)$

$0 = - 2 x y + \frac{\mathrm{dy}}{\mathrm{dx}} \left(- {x}^{2}\right) - \frac{y}{y - 1} + \frac{\mathrm{dy}}{\mathrm{dx}} x \left(\frac{\left(y - 1\right) \left(1\right) - y \left(1\right)}{y - 1} ^ 2\right) + {y}^{2} + \frac{\mathrm{dy}}{\mathrm{dx}} \left(2 y x\right)$

$0 = - 2 x y + \frac{\mathrm{dy}}{\mathrm{dx}} \left(- {x}^{2}\right) - \frac{y}{y - 1} + \frac{\mathrm{dy}}{\mathrm{dx}} x \left(\frac{y - 1 - y}{y - 1} ^ 2\right) + {y}^{2} + \frac{\mathrm{dy}}{\mathrm{dx}} \left(2 y x\right)$

$0 = - 2 x y + \frac{\mathrm{dy}}{\mathrm{dx}} \left(- {x}^{2}\right) - \frac{y}{y - 1} + \frac{\mathrm{dy}}{\mathrm{dx}} x \left(\frac{- 1}{y - 1} ^ 2\right) + {y}^{2} + \frac{\mathrm{dy}}{\mathrm{dx}} \left(2 y x\right)$

$0 = - 2 x y + \frac{\mathrm{dy}}{\mathrm{dx}} \left(- {x}^{2}\right) - \frac{y}{y - 1} + \frac{\mathrm{dy}}{\mathrm{dx}} \frac{- x}{y - 1} ^ 2 + {y}^{2} + \frac{\mathrm{dy}}{\mathrm{dx}} \left(2 y x\right)$

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

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

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