Find the derivative of \(f(x) = x^3 - 4x^2 + 5x - 7\) and evaluate it at \(x = 2\).
["SEO-Optimized Article: Finding the Derivative of (f(x) = x^3 - 4x^2 + 5x - 7) and Evaluating at (x = 2)", "Understanding derivatives is fundamental in calculus, especially when analyzing rates of change, slopes of curves, and optimizing functions. This article walks you through step-by-step how to find the derivative of the function (f(x) = x^3 - 4x^2 + 5x - 7), explains the differentiation rules used, and demonstrates how to evaluate the derivative at (x = 2). Perfect for students, teachers, and math enthusiasts seeking to master basic differentiation.", "---", "### What Is a Derivative?", "The derivative of a function at a point measures how the function’s output changes as its input varies—essentially, the slope of the tangent line to the function’s graph at that point. For polynomial functions like (f(x) = x^3 - 4x^2 + 5x - 7), differentiation is straightforward using standard rules.", "---", "### Step 1: Differentiate Each Term", "Given\n[ f(x) = x^3 - 4x^2 + 5x - 7 ]", "We apply the power rule:\nIf (f(x) = ax^n), then (f'(x) = a \cdot n \cdot x^{n-1}).", "Break it down term by term:", "1. First term: (x^3)\n Derivative: (3x^{3-1} = 3x^2)", "2. Second term: (-4x^2)\n Derivative: (-4 \cdot 2x^{2-1} = -8x)", "3. Third term: (5x)\n Derivative: (5 \cdot 1x^{1-1} = 5)", "4. Constant term: (-7)\n Derivative: (0) (derivatives of constants are always zero)", "---", "### Step 2: Combine the Derivatives", "Add the individual derivatives:\n[ f'(x) = 3x^2 - 8x + 5 ]", "This is the derivative of (f(x)). It represents the instantaneous rate of change of (f(x)) at any point (x).", "---", "### Step 3: Evaluate the Derivative at (x = 2)", "To find the slope of the tangent line at (x = 2), substitute (x = 2) into (f'(x)):", "[\nf'(2) = 3(2)^2 - 8(2) + 5 = 3(4) - 16 + 5 = 12 - 16 + 5 = 1\n]", "---", "### Why Does This Matter?", "Evaluating the derivative at a point gives real-world relevance—such as velocity (rate of change of position), marginal cost in economics, or reaction rates in science. At (x = 2), the function (f(x)) has a slope of 1, meaning the function is increasing at that point with a gentle rise.", "---", "### Summary", "- The derivative of (f(x) = x^3 - 4x^2 + 5x - 7) is:\n [ \boxed{f'(x) = 3x^2 - 8x + 5} ]\n- Evaluated at (x = 2), the derivative is:\n [ \boxed{f'(2) = 1} ]", "Mastering these steps strengthens your calculus foundation and prepares you for more advanced applications. Keep practicing, and use the derivative to unlock deeper insights in math, physics, and engineering!", "---", "Long-tail keywords included for SEO:\n- how to find the derivative of (x^3 - 4x^2 + 5x - 7)\n- derivative of cubic function step-by-step\n- evaluate derivative at (x = 2)\n- find slope of tangent line using differentiation\n- calculus tutorial for beginners", "Meta Description:\nLearn how to differentiate (f(x) = x^3 - 4x^2 + 5x - 7) using the power rule, then evaluate (f'(2) = 1). Step-by-step guide for students and math learners."]
