Let f(x)=

5x
-------
x–2

are we supposed to use the quotient rule? how would we do this. I did the quotient rule and got it wrong.

(x-2)(5)-(5x)(1)
------------------
(x-2)^2

nevermind i got it

To find the derivative of the function f(x) = 5x / (x - 2), we can indeed use the quotient rule. The quotient rule states that if we have a function in the form f(x) = g(x) / h(x), then the derivative of f(x) is given by:

f'(x) = (h(x) * g'(x) - g(x) * h'(x)) / (h(x))^2.

Let's work through the steps to find the derivative of f(x) using the quotient rule:

1. Identify the numerator g(x) and denominator h(x) of the function. In this case, g(x) = 5x, and h(x) = (x - 2).

2. Find the derivative of the numerator g'(x). Since g(x) = 5x, g'(x) = 5.

3. Find the derivative of the denominator h'(x). In this case, h(x) = (x - 2), so h'(x) = 1.

4. Apply the quotient rule formula:

f'(x) = (h(x) * g'(x) - g(x) * h'(x)) / (h(x))^2

= [(x - 2) * 5 - 5x * 1] / (x - 2)^2

= [5x - 10 - 5x] / (x - 2)^2

= -10 / (x - 2)^2.

So, the derivative of f(x) is f'(x) = -10 / (x - 2)^2.

Please note that if you obtained a different result, it could be due to a calculation error, so you may want to double-check your work.