Question

How do you differentiate `f(x)=(x+3)^2/(x-1)` using the quotient rule?

Answer 1

You have to first differentiate the numerator using the chain rule.

Explanation:

Let `y = u^2` and `u = x + 3`. Then `y' = 2u` and #u' = 1. The derivative of the numerator is :

`dy/dx = 2u xx 1 = 2(x + 3) = 2x + 6`

Now, we can use the quotient rule to differentiate the entire function.

`f'(x) = ((2x + 6)(x - 1) - 1(x + 3)^2)/(x - 1)^2`

`f'(x) = (2x^2 + 6x - 2x - 6 - 1(x^2 + 6x + 9))/(x^2 - 2x + 1)`

`f'(x) = (x^2 - 2x -15)/(x^2 - 2x + 1)`

`f'(x) = ((x - 5)(x + 3))/(x - 1)^2`

Hopefully this helps!