# How do you differentiate y^2-y+5=3x^2-6x+3?

Dec 14, 2015

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{6 \left(x - 1\right)}{2 y - 1}$

#### Explanation:

Implicit differentiation is basically glorified chain rule. Whenever you differentiate a term containing $y$ with respect to $x$, a $\frac{\mathrm{dy}}{\mathrm{dx}}$ term will be spit out.

$\frac{d}{\mathrm{dx}} \left[{y}^{2} - y + 5 = 3 {x}^{2} - 6 x + 3\right]$

$2 y \frac{\mathrm{dy}}{\mathrm{dx}} - \frac{\mathrm{dy}}{\mathrm{dx}} = 6 x - 6$

$\frac{\mathrm{dy}}{\mathrm{dx}} \left(2 y - 1\right) = 6 \left(x - 1\right)$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{6 \left(x - 1\right)}{2 y - 1}$