Question

How do you differentiate `f(x)= (x^2-x+2)/ (x- 7 )` using the quotient rule?

Answer 1

`f'(x)=1-44/(x-7)^2`

Explanation:

The quotient rule states that
`d/dx(u/v)=(u'v-uv')/v^2`

In our case, we have
`u=x^2-x+2->u'=2x-1`
`v=x-7->v'=1`

Applying the power rule and doing some algebra,
`f'(x)=((x^2-x+2)'(x-7)-(x^2-x+2)(x-7)')/(x-7)^2`
`color(white)(XX)=((2x-1)(x-7)-(x^2-x+2))/(x-7)^2`
`color(white)(XX)=(2x^2-15x+7-x^2+x-2)/(x-7)^2`
`color(white)(XX)=(x^2-14x+5)/(x-7)^2`
`color(white)(XX)=(x^2-14x+49-44)/(x-7)^2`
`color(white)(XX)=((x-7)^2-44)/(x-7)^2`
`color(white)(XX)=(x-7)^2/(x-7)^2-44/(x-7)^2`
`color(white)(XX)=1-44/(x-7)^2`