Question

How do you use the quotient rule to differentiate `f(x)= (2x^2-x+1)/(2x-1)`?

Answer 1

`f'(x)=(4x^2-4x-1)/(2x-1)^2`

Explanation:

`"Given " f(x)=(g(x))/(h(x))" then "`

`color(red)(bar(ul(|color(white)(2/2)color(black)(f'(x)=(h(x)g'(x)-g(x)h'(x))/(h(x))^2)color(white)(2/2)|)))`

`"here " g(x)=2x^2-x+1rArrg'(x)=4x-1`

`"and " h(x)=2x-1rArrh'(x)=2`

`rArrf'(x)=((2x-1)(4x-1)-(2x^2-x+1).2)/(2x-1)^2`

simplifying the numerator.

`f'(x)=(8x^2-6x+1-4x^2+2x-2)/(2x-1)^2`

`=(4x^2-4x-1)/(2x-1)^2`