f'(x) = rac{d}{dx}(3x^3 - 5x^2 + 2x - 7) = 9x^2 - 10x + 2

["# Understanding the Derivative: f′(x) = (3x³ − 5x² + 2x − 7)’ = 9x² − 10x + 2", "When studying calculus, one of the most fundamental operations you’ll encounter is differentiation—the process of finding the derivative of a function. Today, we’ll explore the derivative of a cubic polynomial and clarify why the result is ( f'(x) = 9x^2 - 10x + 2 ). Whether you're a student learning calculus or a self-learner brushing up on key concepts, this article breaks down the derivative computation step-by-step.", "---", "## What Is a Derivative?", "The derivative of a function at a point represents its instantaneous rate of change or slope of the tangent line at that point. In more practical terms, it tells us how a function’s value changes as its input changes—crucial in physics, economics, engineering, and virtually every quantitative field.", "---", "## The Function: f(x) = 3x³ − 5x² + 2x − 7", "We begin with the cubic polynomial:", "[\nf(x) = 3x^3 - 5x^2 + 2x - 7\n]", "This expression consists of four terms representing different polynomial degrees:", "- A cubic term: ( 3x^3 )\n- A quadratic term: ( -5x^2 )\n- A linear term: ( +2x )\n- A constant term: ( -7 )", "Differentiating each component individually is the key to solving the derivative.", "---", "## Step-by-Step Differentiation", "Using the power rule—which states that if ( f(x) = ax^n ), then ( f'(x) = a \cdot n \cdot x^{n-1} )—we compute the derivative term by term:", "### 1. Differentiate ( 3x^3 ):", "[\n\frac{d}{dx}(3x^3) = 3 \cdot 3 \cdot x^{3-1} = 9x^2\n]", "### 2. Differentiate ( -5x^2 ):", "[\n\frac{d}{dx}(-5x^2) = -5 \cdot 2 \cdot x^{2-1} = -10x\n]", "### 3. Differentiate ( 2x ):", "[\n\frac{d}{dx}(2x) = 2 \cdot 1 \cdot x^{1-1} = 2x^0 = 2\n]", "(Note: The derivative of 1 is 1, so ( 2x ) becomes a constant: 2.)", "### 4. Differentiate ( -7 ):", "[\n\frac{d}{dx}(-7) = 0 \quad \ ext{(derivative of any constant is 0)}\n]", "---", "## Combine All Terms", "Now, sum the derivatives of each individual term:", "[\nf'(x) = 9x^2 + (-10x) + 2 + 0 = 9x^2 - 10x + 2\n]", "---", "## Final Result", "Thus, the derivative of ( f(x) = 3x^3 - 5x^2 + 2x - 7 ) is:", "[\n\boxed{f'(x) = 9x^2 - 10x + 2}\n]", "This expression helps us analyze the function’s behavior—finding maxima, minima, or rates of increase and decrease—making it indispensable in both pure and applied mathematics.", "---", "## Why This Matters", "Understanding how to differentiate polynomials allows you to solve real-world problems, from optimizing profit functions to modeling motion. The rule ( \frac{d}{dx}(ax^n) = anx^{n-1} ) forms the backbone for differentiating more complex expressions, especially when combined with summation rules and product/quotient differentiation.", "---", "## Summary", "- The derivative ( f'(x) = 9x^2 - 10x + 2 ) comes from applying the power rule term-by-term to the original cubic function\n- Polynomial differentiation is efficient and systematic\n- The result describes the rate of change of ( f(x) ) at any point ( x )", "Whether you’re seeing this in a classroom, on an exam, or independently, mastering the derivative of cubic functions sets a strong foundation in calculus.", "---", "Keywords: derivative, f prime, f′(x), calculus tutorial, differentiation example, power rule, polynomial derivative, 9x² − 10x + 2, how to find f’(x), derivative of 3x³ − 5x² + 2x − 7", "---", "References & Further Reading:", "- Stewart, James. Calculus: Early Transcendentals\n- Khan Academy: Introduction to Derivatives\n- Paul’s Online Math Notes: Differentiation", "---", "Stay tangent with calculus—every derivative reveals a deeper truth about change."]