Problem
Find the difference quotient for the function \(f(x) = 3x^2 - 2x + 1\).
Solution
Step 1 ：The difference quotient formula is \(\frac{f(x + h) - f(x)}{h}\).
Step 2 ：First, we compute \(f(x + h) = 3(x + h)^2 - 2(x + h) + 1 = 3x^2 + 6hx + 3h^2 - 2x - 2h + 1\).
Step 3 ：Then, we substitute \(f(x + h)\) and \(f(x)\) into the formula, giving \(\frac{3x^2 + 6hx + 3h^2 - 2x - 2h + 1 - (3x^2 - 2x + 1)}{h} = \frac{6hx + 3h^2 - 2h}{h} = 6x + 3h - 2\).
