Question:** If \(f(x) = 3x^2 - 2x + 1\), find \(f'(x)\) and evaluate \(f'(2)\).
["Understanding the Derivative: How to Find (f'(x)) and Evaluate (f'(2)) for (f(x) = 3x^2 - 2x + 1)", "When studying calculus, one of the most fundamental and essential tasks is finding the derivative of a function. The derivative represents the rate of change of a function and opens the door to deeper insights in optimization, motion analysis, and more. In this article, we dive into the specific problem:\nIf (f(x) = 3x^2 - 2x + 1), find (f'(x)) and evaluate (f'(2)).", "---", "### Step 1: Find the derivative (f'(x))", "To differentiate the function (f(x) = 3x^2 - 2x + 1), we apply standard rules of differentiation.", "The function is a polynomial, so we use:\n- The power rule: (\frac{d}{dx}[x^n] = nx^{n-1})\n- Constants multiplied by functions: (\frac{d}{dx}[c \cdot g(x)] = c \cdot g'(x))\n- Sum/difference rule: (\frac{d}{dx}[g(x) \pm h(x)] = g'(x) \pm h'(x))", "Let’s differentiate each term:", "1. The derivative of (3x^2) is (3 \cdot 2x^{2-1} = 6x)\n2. The derivative of (-2x) is (-2 \cdot 1x^{1-1} = -2)\n3. The derivative of the constant (+1) is (0)", "Putting it together:\n[\nf'(x) = 6x - 2\n]", "---", "### Step 2: Evaluate (f'(2))", "Now that we have (f'(x) = 6x - 2), substitute (x = 2):\n[\nf'(2) = 6(2) - 2 = 12 - 2 = 10\n]", "---", "### Why This Matters: The Applications of (f'(x))", "Finding the derivative (f'(x) = 6x - 2) helps understand how the function changes. Evaluating (f'(2) = 10) tells us the slope of the tangent line to the curve (f(x)) at (x = 2), which is crucial in fields like physics (velocity), economics (marginal cost), and engineering (optimization).", "---", "### Summary", "- Given (f(x) = 3x^2 - 2x + 1),\n[\nf'(x) = 6x - 2\n]\n- Evaluating at (x = 2):\n[\nf'(2) = 10\n]", "This simple yet powerful example demonstrates the core principles of differentiation and showcases how derivatives quantify change — a cornerstone concept in calculus and its applications.", "---", "### SEO Keywords & Phrases\n- find (f'(x)) for (f(x) = 3x^2 - 2x + 1)\n- how to differentiate quadratic functions\n- evaluate (f'(2)) for polynomial function\n- understanding the derivative and its applications\n- step-by-step derivative calculation and evaluation", "---", "Start mastering your calculus skills today — derivatives are not just formulas, but tools to unlock real-world insights!"]
