Question

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

Answer 1

As they share the same denominator, we can add those fractions and then apply quotient rule to the function (now aggregated): `f(x)=(x-2)/(2x-2)`

Explanation:

Quotient rule states that for `y=f(x)/g(x)`, then `(dy)/(dx)=(f'(x)g(x)-f(x)g'(x))/g(x)^2`

So:

`(dy)/(dx)=(1(2x-2)-(x-2)(2))/(2x-2)^2=(2x-2-2x+4)/(2x-2)^2`

`(dy)/(dx)=2/(2x-2)^2=2/(4x^2-4x+4)=1/(2(x^2-x+1))`