Question

How do you differentiate `f(x)=(x-5)/(x-3)^2` using the quotient rule?

Answer 1

`f'(x)=(-x+7)/(x-3)^3 = -(x-7)/(x-3)^3`

Explanation:

How to differentiate using the quotient rule?

Given `f(x)= (x-5) /(x-3)^2`

`f'(x)= ((x-3)^2 d/dx(x-5) - (x-5)d/dx(x-3)^2)/((x-3)^2)^2`

`=((x-3)^2(1) -(x-5)(2(x-3)(1)))/(x-3)^4`

`=(color(red)((x-3))(x-3) -2color(red)((x-3))(x-5))/(color(red)((x-3))(x-3)^3)`

Factor out the greatest common factor `(x-3)`

`f'(x) = ((x-3)-2(x-5))/(x-3)^3`

`= (x-3-2x+10)/(x-3)^3`

`f'(x)=(-x+7)/(x-3)^3 = -(x-7)/(x-3)^3`