AP Calculus AB Exam Question:

Let f(x) = 3x^2 - 4x. Find the derivative of f(x) using the definition of the derivative.

Answer with Step-by-Step Explanation:

To find the derivative of f(x) using the definition of the derivative, we need to use the formula:

f'(x) = lim(h -> 0) [f(x + h) - f(x)] / h

Step 1: Substitute the function f(x) = 3x^2 - 4x into the formula.

f'(x) = lim(h -> 0) [(3(x + h)^2 - 4(x + h)) - (3x^2 - 4x)] / h

Step 2: Expand the expressions within the brackets.

f'(x) = lim(h -> 0) [(3(x^2 + 2xh + h^2) - 4(x + h)) - (3x^2 - 4x)] / h

Step 3: Simplify by distributing the coefficients.

f'(x) = lim(h -> 0) [3x^2 + 6xh + 3h^2 - 4x - 4h - 3x^2 + 4x] / h

Step 4: Combine like terms.

f'(x) = lim(h -> 0) [6xh + 3h^2 - 4h] / h

Step 5: Factor out h from the numerator.

f'(x) = lim(h -> 0) h(6x + 3h - 4) / h

Step 6: Cancel the h terms.

f'(x) = lim(h -> 0) 6x + 3h - 4

Step 7: Take the limit as h approaches 0.

f'(x) = 6x + 0 - 4

Step 8: Simplify.

f'(x) = 6x - 4

Therefore, the derivative of f(x) = 3x^2 - 4x using the definition of the derivative is f'(x) = 6x - 4.