["Title: Comprehensive Guide to the Derivative of (12x^2 - 10x + 2)", "---", "### Understanding the Derivative of (12x^2 - 10x + 2)", "Learning how to find the derivative of polynomial functions is essential in calculus, and mastering the derivative of (12x^2 - 10x + 2) is a foundational step. This guide breaks down the process step-by-step, explains its importance, and explores real-world applications.", "---", "### What is a Derivative?", "In calculus, the derivative of a function represents the rate at which the function’s value changes at any given point. Geometrically, it corresponds to the slope of the tangent line to the function’s graph at that point. For polynomial functions like (12x^2 - 10x + 2), derivatives are straightforward to compute.", "---", "### Step-by-Step Derivative of (12x^2 - 10x + 2)", "Given the function:", "[
\nf(x) = 12x^2 - 10x + 2
\n]", "To find (f'(x)), apply basic differentiation rules:", "1. Power Rule:
\n The derivative of (x^n) is (n x^{n-1}).
\n - For (12x^2): (2 \cdot 12x^{2-1} = 24x)
\n - For (-10x): (1 \cdot (-10)x^{1-1} = -10)
\n - For the constant (2): The derivative of any constant is (0).", "2. Combine all terms:", "[
\nf'(x) = 24x - 10 + 0 = 24x - 10
\n]", "---", "### Why Derivatives Matter: Key Applications", "- Finding Slopes: The derivative (24x - 10) gives the slope of the original curve at any (x), helping analyze increasing/decreasing behavior.
\n- Critical Points & Extrema: Setting (f'(x) = 0):
\n [
\n 24x - 10 = 0 \implies x = \frac{10}{24} = \frac{5}{12}
\n ]
\n This critical point indicates potential maxima or minima.
\n- Optimization: Useful in business and engineering for maximizing profits or minimizing costs where the function represents yield or cost.
\n- Motion Analysis: In physics, derivatives represent velocity (derivative of position) and acceleration.", "---", "### Graph Behavior and Derivative Insights", "The original function (12x^2 - 10x + 2) is a parabola opening upwards (since the coefficient of (x^2) is positive). Its derivative (24x - 10) is a straight line:
\n- Negative for (x < \frac{5}{12}) → Function decreasing
\n- Zero at (x = \frac{5}{12}) → tangent horizontal (minimum point)
\n- Positive for (x > \frac{5}{12}) → Function increasing", "This confirms the vertex of the parabola occurs at (x = \frac{5}{12}), matching known parabola properties.", "---", "### Summary", "The derivative of (12x^2 - 10x + 2) is:", "[
\n\boxed{f'(x) = 24x - 10}
\n]", "Understanding derivatives opens doors to deeper calculus concepts and practical tools in science, economics, and engineering. Whether analyzing curves or modeling real-world phenomena, this simple quadratic’s derivative illustrates core principles vital for further study.", "---", "### Further Reading", "- Learn about the Power Rule in depth
\n- Explore Applications of Derivatives in Physics
\n- Master Chain Rule for composite functions
\n- Take online quizzes on polynomial differentiation", "---", "Keywords: derivative of (12x^2 - 10x + 2), f'(x) = 24x - 10, calculus tutorial, polynomial differentiation, rate of change, optimization, tangent line, calculus basics."]
