Question

How do you find the derivative of `(x^2-4)/(x-1)`?

Answer 1

`d/(dx) ((x^2-4)/(x-1)) = 1+3/(x-1)^2`

Explanation:

I would simplify the expression first, then use the power rule...

`d/(dx) ((x^2-4)/(x-1)) = d/(dx) ((x^2-x+x-1-3)/(x-1))`

`color(white)(d/(dx) ((x^2-4)/(x-1))) = d/(dx) (((x+1)(x-1)-3)/(x-1))`

`color(white)(d/(dx) ((x^2-4)/(x-1))) = d/(dx) ((x+1)-3/(x-1))`

`color(white)(d/(dx) ((x^2-4)/(x-1))) = d/(dx) (x+1-3(x-1)^(-1))`

`color(white)(d/(dx) ((x^2-4)/(x-1))) = 1+0+3(x-1)^(-2)`

`color(white)(d/(dx) ((x^2-4)/(x-1))) = 1+3/(x-1)^2`

`color(white)()`
Note: I did use the chain rule quietly above too, when differentiating:

`(x-1)^(-1)`

to get:

`-1(x-1)^(-2)*d/(dx)(x-1) = -(x-1)^(-2)`