# How do you differentiate f(x)=x/(2x-2)-2/(2x-2) using the quotient rule?

Jan 1, 2016

As they share the same denominator, we can add those fractions and then apply quotient rule to the function (now aggregated): $f \left(x\right) = \frac{x - 2}{2 x - 2}$

#### Explanation:

Quotient rule states that for $y = f \frac{x}{g} \left(x\right)$, then $\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{f ' \left(x\right) g \left(x\right) - f \left(x\right) g ' \left(x\right)}{g} {\left(x\right)}^{2}$

So:

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{1 \left(2 x - 2\right) - \left(x - 2\right) \left(2\right)}{2 x - 2} ^ 2 = \frac{2 x - 2 - 2 x + 4}{2 x - 2} ^ 2$

$\frac{\mathrm{dy}}{\mathrm{dx}} = \frac{2}{2 x - 2} ^ 2 = \frac{2}{4 {x}^{2} - 4 x + 4} = \frac{1}{2 \left({x}^{2} - x + 1\right)}$