Question: The function f(x) is defined as f(x) = 3x^2 - 4x + 2. Find the derivative of the function f(x) using first principles. Show all your work.

Answer: To find the derivative of the function f(x) using first principles, we need to use the definition of the derivative:

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

First, we need to find f(x + h) using the given function f(x) = 3x^2 - 4x + 2: f(x + h) = 3(x + h)^2 - 4(x + h) + 2 = 3(x^2 + 2hx + h^2) - 4x - 4h + 2 = 3x^2 + 6hx + 3h^2 - 4x - 4h + 2

Next, we can substitute f(x) = 3x^2 - 4x + 2 and f(x + h) = 3x^2 + 6hx + 3h^2 - 4x - 4h + 2 into the definition of the derivative:

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

So, the derivative of the function f(x) = 3x^2 - 4x + 2 using first principles is f'(x) = 6x - 4.