Question

How do you differentiate `f(x)=(2x^2-3x+2)/(x-2)` using the quotient rule?

Answer 1

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

Explanation:

The Quotient Rule tells us that
if `f(x)=(g(x))/(h(x))`
then
`color(white)("XXX")f'(x)=(g'(x) * h(x) - g(x) * h'(x))/(h^2(x))`

Since we are told `f(x)=(2x^2-3x+2)/(x-2)`
it would seem reasonable to let
`color(white)("XXX")g(x)=2x^2-3x+2`
and
`color(white)("XXX")h(x)=x-2`

Using the exponent rule for derivatives, this implies
`color(white)("XXX")g'(x)=4x-3`
and
`color(white)("XXX")h'(x)=1`

Furthermore, by simple multiplication:
`color(white)("XXX")h^2(x)=x^2-4x+4`

Evaluating one term at a time for the numerator:
#{: (g'(x) * h(x)=,color(white)("xxx"),4x^2-11x+6), (ul(g(x) * h'(x)=),,ul(2x^2-3x+2)), (g'(x) * h(x) - g(x) * h'(x)=,,2x^2-8x+4) :}#

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