Find the derivative of \(f(x) = 3x^3 - 5x^2 + 6x - 4\).
["# Find the Derivative of (f(x) = 3x^3 - 5x^2 + 6x - 4)", "Understanding the derivative of a polynomial function is fundamental in calculus, especially when analyzing the rate of change, optimization, or curve sketching. In this article, we’ll find the derivative of the function\n[ f(x) = 3x^3 - 5x^2 + 6x - 4 ]\nusing standard differentiation rules.", "---", "## What is a Derivative?", "In calculus, the derivative of a function at a point represents the slope of the tangent line to the function’s graph at that point. It also gives information about how the function is increasing or decreasing.", "---", "## Differentiating Term by Term", "The function\n[ f(x) = 3x^3 - 5x^2 + 6x - 4 ]\nis a polynomial composed of four terms. We apply the Basic Differentiation Rules:", "1. Power Rule: If ( f(x) = ax^n ), then ( f'(x) = a \cdot n x^{n-1} )\n2. Constant Rule: The derivative of any constant is 0\n3. Linearity: Derivative of a sum/difference is the sum/difference of the derivatives", "Let’s differentiate each term:", "- For (3x^3):\n [ \frac{d}{dx}(3x^3) = 3 \cdot 3x^{3-1} = 9x^2 ]", "- For (-5x^2):\n [ \frac{d}{dx}(-5x^2) = -5 \cdot 2x^{2-1} = -10x ]", "- For (6x):\n [ \frac{d}{dx}(6x) = 6 \cdot 1x^{1-1} = 6 ]", "- For (-4):\n [ \frac{d}{dx}(-4) = 0 ] (constant)", "---", "## Combine the Derivatives", "Now, sum up all the derivatives:", "[\nf'(x) = 9x^2 - 10x + 6 + 0 = 9x^2 - 10x + 6\n]", "---", "## Final Answer", "[\n\boxed{f'(x) = 9x^2 - 10x + 6}\n]", "---", "## Why This Derivative Matters", "- Rate of Change: (f'(x)) tells us how fast (f(x)) changes at every point.\n- Critical Points: Setting (f'(x) = 0) helps identify local maxima, minima, and inflection points.\n- Applications: Used in physics (velocity is derivative of position), economics (marginal cost), and engineering.", "---", "## Summary", "To find the derivative of (f(x) = 3x^3 - 5x^2 + 6x - 4), apply the power rule to each term and combine the results:", "[\n\frac{d}{dx}(3x^3 - 5x^2 + 6x - 4) = 9x^2 - 10x + 6\n]", "Mastering derivative rules like these forms the backbone of advanced calculus and real-world problem solving.", "---", "Keywords: derivative of (3x^3 - 5x^2 + 6x - 4), find f’(x), differentiation rules, calculus, tangent slope, polynomial derivative, mathematics tutorial.\nMeta Description: Learn how to find the derivative of (f(x) = 3x^3 - 5x^2 + 6x - 4) using the power rule and basic differentiation techniques. Step-by-step explanation with full derivation."]
