AP Calculus AB Exam Question:

Find the derivative of the function f(x) = (3x + 2) / (2x^2 - 5x + 1).

Step-by-step Solution:

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

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

Let's begin by identifying the functions in our problem:

u(x) = 3x + 2

v(x) = 2x^2 - 5x + 1

Next, we need to find the derivatives of u(x) and v(x):

u'(x) = 3 (derivative of 3x + 2 with respect to x)

v'(x) = 4x - 5 (derivative of 2x^2 - 5x + 1 with respect to x)

Substituting these values into the quotient rule, we get:

f'(x) = [(3)(2x^2 - 5x + 1) - (3x + 2)(4x - 5)] / [2x^2 - 5x + 1]^2

Simplifying further:

f'(x) = (6x^2 - 15x + 3 - (12x^2 - 35x - 8x + 10)) / [2x^2 - 5x + 1]^2

f'(x) = (6x^2 - 15x + 3 - 12x^2 + 35x - 8x + 10) / [2x^2 - 5x + 1]^2

f'(x) = (-6x^2 + 12x + 13) / [2x^2 - 5x + 1]^2

Therefore, the derivative of the function f(x) = (3x + 2) / (2x^2 - 5x + 1) is f'(x) = (-6x^2 + 12x + 13) / [2x^2 - 5x + 1]^2.