AP Calculus AB Exam Question:

Find the derivative of the function f(x) = 3x^4 - 2x^2 + 5x - 1.

Step-by-step Solution:

To find the derivative of the given function, we will use the power rule and the sum/difference rule of differentiation. The power rule states that if f(x) = x^n, then f'(x) = nx^(n-1).

Let's differentiate term by term:

f(x) = 3x^4 - 2x^2 + 5x - 1

To differentiate 3x^4, we use the power rule. The derivative of 3x^4 with respect to x gives: f'(x) = (3)(4)(x^3) = 12x^3

Next, we differentiate -2x^2: f'(x) = 12x^3 - (2)(2)(x^1) = 12x^3 - 4x

Moving on to 5x, its derivative is: f'(x) = 12x^3 - 4x + 5

Since the derivative of a constant is zero, the last term -1 does not contribute to the derivative, so we can ignore it.

The final derivative of the function f(x) = 3x^4 - 2x^2 + 5x - 1 is: f'(x) = 12x^3 - 4x + 5

Hence, the derivative of the given function is 12x^3 - 4x + 5.