The derivative of f(x) = 3x⁴ − 5x² + 2x is:

["# The Derivative of f(x) = 3x⁴ − 5x² + 2x: Everything You Need to Know", "Understanding derivatives is fundamental in calculus, especially when analyzing the rate of change of functions — powerful tools used in physics, economics, engineering, and data science. In this article, we dive into the derivative of the function f(x) = 3x⁴ − 5x² + 2x, explaining not only the computation but also its significance and applications.", "## What is a Derivative?", "Before we compute the derivative, let’s recall what it means. The derivative of a function at a point represents the instantaneous rate of change of the function at that point — essentially, the slope of the function’s graph. In mathematical terms, if ( f(x) ) is a differentiable function, the derivative is written as ( f'(x) ) or ( \frac{df}{dx} ).", "## Computing the Derivative: Step-by-Step", "Given the function:\n[ f(x) = 3x^4 - 5x^2 + 2x ]", "We apply standard rules of differentiation, including:", "- Power Rule: ( \frac{d}{dx}[x^n] = nx^{n-1} )\n- Constant Multiplier Rule: ( \frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x) )\n- Sum Rule: ( \frac{d}{dx}[f(x) + g(x)] = f'(x) + g'(x) )", "Let’s differentiate each term:", "1. First term: ( 3x^4 )\n Applying the power rule:\n [\n \frac{d}{dx}[3x^4] = 3 \cdot 4x^{3} = 12x^3\n ]", "2. Second term: ( -5x^2 )\n [\n \frac{d}{dx}[-5x^2] = -5 \cdot 2x = -10x\n ]", "3. Third term: ( 2x )\n [\n \frac{d}{dx}[2x] = 2 \cdot 1x^0 = 2\n ]", "Now, combining all the derivatives using the sum rule:\n[\nf'(x) = 12x^3 - 10x + 2\n]", "## Final Answer", "[\n\boxed{f'(x) = 12x^3 - 10x + 2}\n]", "## Why This Derivative Matters", "Knowing that ( f'(x) = 12x^3 - 10x + 2 ) allows you to:", "- Find critical points: Set ( f'(x) = 0 ) to determine maxima, minima, or points of inflection — key in optimization problems.\n- Analyze function behavior: The sign of ( f'(x) ) tells you where the function is increasing or decreasing.\n- Solve real-world problems: In physics, derivatives describe velocity (derivative of position) and acceleration (derivative of velocity). For example, if ( f(x) ) models displacement over time, ( f'(x) ) is instantaneous velocity.", "## Visual Guide: Linking the Derivative to the Graph", "Graphically, the graph of ( f'(x) = 12x^3 - 10x + 2 ) shows how the slope of ( f(x) ) changes across the domain. Local peaks and valleys in ( f'(x) ) correspond to inflection points in ( f(x) ), while where ( f'(x) = 0 ) indicates horizontal tangents (critical points).", "## Conclusion", "The derivative of ( f(x) = 3x^4 − 5x² + 2x ) is ( f'(x) = 12x^3 - 10x + 2 ). This result is not just a mathematical formula — it’s a vital tool for understanding how functions behave, enabling deeper insights into patterns, trends, and optimization across countless disciplines. Whether you're a student, educator, or professional in STEM fields, mastering derivatives empowers you to interpret change with precision and confidence.", "---", "### Key SEO Keywords to Optimize This Article\n- The derivative of f(x) = 3x⁴ − 5x² + 2x\n- Derivative of 3x⁴ − 5x² + 2x explained\n- Calculus derivative guide\n- How to find f'(x)\n- Applications of derivatives\n- Learn calculus derivatives", "---", "Call to Action: Struggling with derivatives? Practice identifying derivatives of polynomial functions and explore their applications using tools like graphing calculators and calculus software. Keep learning — the deeper your understanding, the more powerful your analytical skills become!"]