AP Calculus AB Exam Question:

Let f(x) = (x^2 + 3x - 2) / (x - 1). Using the quotient rule, find the derivative of f(x).

Answer:

To find the derivative of f(x), we will use the quotient rule, which states that for any two functions u(x) and v(x), the derivative of their quotient u(x) / v(x) is given by:

(u'(x) * v(x) - v'(x) * u(x)) / (v(x))^2

Let's apply this rule to our given function:

Given: f(x) = (x^2 + 3x - 2) / (x - 1)

Let u(x) = x^2 + 3x - 2, v(x) = x - 1.

We first need to find u'(x) and v'(x).

Differentiating u(x), we get: u'(x) = 2x + 3

Differentiating v(x), we get: v'(x) = 1

Now, using the quotient rule, the derivative of f(x) is given by:

f'(x) = (u'(x) * v(x) - v'(x) * u(x)) / (v(x))^2

= [(2x + 3) * (x - 1) - 1 * (x^2 + 3x - 2)] / (x - 1)^2

= [2x^2 + 3x - 2x - 3 - x^2 - 3x + 2] / (x - 1)^2

= (x^2 - x - 1) / (x - 1)^2

Thus, the derivative of f(x) is f'(x) = (x^2 - x - 1) / (x - 1)^2.

Note: It is always a good practice to simplify the final expression if possible.