First, find the derivative: \( f'(x) = 3x^2 - 6x \).

["# First, Find the Derivative: ( f'(x) = 3x^2 - 6x )", "Understanding derivatives is a fundamental concept in calculus, with wide applications in science, engineering, economics, and beyond. One of the first steps in analyzing functions mathematically is finding their derivatives. If you’re asked to first find the derivative ( f'(x) = 3x^2 - 6x ), this article will guide you through the process step-by-step—whether you’re brand new to derivatives or refreshing your math skills.", "## Why Learn Derivatives?", "Before diving into calculations, it helps to understand why derivatives matter. The derivative of a function at a point represents the slope of the tangent line to its graph at that point. This concept is central to optimization, motion analysis, and changes in real-world systems—making early mastery crucial.", "---", "## Step 1: Recall the Power Rule", "The most common rule for differentiating polynomial functions is the Power Rule. It states:", "> If ( f(x) = ax^n ), then ( f'(x) = a \cdot n x^{n-1} ).", "In simpler terms, multiply by the exponent and reduce the exponent by one.", "---", "## Step 2: Differentiate Each Term", "The given function is:", "[\nf(x) = 3x^2 - 6x\n]", "We apply the Power Rule to each term separately.", "### Differentiate ( 3x^2 ):", "- Coefficient: 3\n- Exponent: 2\n- Applying the rule: ( 3 \cdot 2 x^{2-1} = 6x )", "### Differentiate ( -6x ):\nRecall that ( x = x^1 ), so\n- Exponent: 1\n- Applying the rule: ( -6 \cdot 1 x^{1-1} = -6x^0 = -6 \cdot 1 = -6 )", "---", "## Step 3: Combine the Results", "Add the derivatives of both terms:", "[\nf'(x) = 6x - 6\n]", "Wait—there’s a small adjustment to ensure consistency with the original expression ( f'(x) = 3x^2 - 6x ). However, note: the derivative derived is ( 6x - 6 ), which actually simplifies back to ( -6 + 6x ). So while technically correct, expressing it as ( f'(x) = 6x - 6 ) reflects the same function in standard linear form.", "Indeed, observing:", "[\nf'(x) = 6x - 6 = 3(2x^2 - 2x) - \ ext{corresponding constant shift}\n]", "But hold—this reveals a key insight: frequently simplifying derivatives into factored or expanded forms improves intuition and usability.", "Indeed, factoring gives:", "[\nf'(x) = 3(x^2 - 2x)\n]", "But since the original expression was ( f'(x) = 3x^2 - 6x ), the derivative ( 6x - 6 ) is also accurate—though convention favors expressions matching the original degree pattern.", "---", "## Final Answer (Most Standard Form)", "[\n\boxed{f'(x) = 6x - 6}\n]", "It’s important to confirm what form matches expectations. The original function ( f(x) = 3x^2 - 6x ) is quadratic, so its derivative must be linear. Therefore, ( f'(x) = 6x - 6 ) is correct and consistent.", "---", "## Practice Tip", "If you’re learning this, try differentiating similar quadratics:", "- ( f(x) = 5x^2 - 3x \Rightarrow f'(x) = 10x - 3 )\n- ( f(x) = -4x^2 + 7x - 2 \Rightarrow f'(x) = -8x + 7 )", "---", "## Conclusion", "Finding the derivative ( f'(x) = 3x^2 - 6x )’s derivation hinges on applying the Power Rule correctly to each term. Recall that derivatives contribute linear changes—ensuring your result reflects powered, reduced-degree terms are key. Although the derivative can be expressed as ( 6x - 6 ), verifying consistency with the original function’s degree confirms the accuracy of your answer.", "Whether you’re learning calculus for school, engineering, or personal growth, mastering differentiation starts with building a strong foundation in rule-based computation—and this derivation is a perfect first step.", "---", "Key Search Terms for SEO:\n- How to find the derivative of ( 3x^2 - 6x )\n- First derivative step-by-step\n- Derivative of quadratic function\n- Power rule differentiation\n- Calculus fundamentals for beginners"]