Find the derivative of the function \(f(x) = 3x^3 - 5x^2 + 6x - 4\).

["# Find the Derivative of the Function (f(x) = 3x^3 - 5x^2 + 6x - 4) — A Step-by-Step Guide", "Understanding how to find the derivative of a function is fundamental in calculus, especially when analyzing rates of change and function behavior. In this comprehensive guide, we’ll walk through how to find the derivative of the cubic polynomial function:", "[\nf(x) = 3x^3 - 5x^2 + 6x - 4\n]", "If you're asking: “Find the derivative of (f(x) = 3x^3 - 5x^2 + 6x - 4)”, this article provides a clear, step-by-step solution with explanations to help you master differentiation techniques.", "---", "## Why Find Derivatives?", "Derivatives provide critical insights into the behavior of functions:", "- The derivative (f'(x)) represents the slope of the tangent to the curve at any point (x).\n- Derivatives help locate maxima, minima, and inflection points.\n- They are essential in optimization problems, motion analysis, economics, physics, and many more.", "---", "## Step-by-Step Differentiation", "The function (f(x) = 3x^3 - 5x^2 + 6x - 4) is a polynomial, and ideal for applying the power rule of differentiation. For any term (ax^n), the derivative is:", "[\n\frac{d}{dx}[ax^n] = a \cdot n \cdot x^{n-1}\n]", "Let’s differentiate each term individually.", "### 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 (6x)", "[\n\frac{d}{dx}[6x] = 6 \cdot 1 \cdot x^{1-1} = 6x^0 = 6\n]", "(Note: The derivative of (x) is 1, and any constant (like 6) becomes 0.)", "### 4. Differentiate (-4)", "[\n\frac{d}{dx}[-4] = 0\n]", "Constants vanish under differentiation.", "---", "## Combine All Terms", "Now, sum the derivatives of each term:", "[\nf'(x) = 9x^2 - 10x + 6 + 0 = 9x^2 - 10x + 6\n]", "---", "## Final Result", "[\n\boxed{f'(x) = 9x^2 - 10x + 6}\n]", "This is the derivative of the function (f(x) = 3x^3 - 5x^2 + 6x - 4).", "---", "## Summary", "To find the derivative of a polynomial:", "1. Apply the power rule to each term: multiply by the exponent, then reduce the exponent by 1.\n2. Simplify constant terms.\n3. Combine all resulting terms into the derivative.", "Understanding this process not only solves the problem at hand but strengthens your foundation in calculus, opening doors to advanced topics like related rates, curve sketching, and optimization.", "For further practice, try differentiating higher-degree polynomials and explore the derivative’s meaning in real-world applications like velocity or marginal cost.", "---", "Keywords: derivative of (3x^3 - 5x^2 + 6x - 4), find derivative, differentiation rules, calculus tutorial, power rule, polynomial derivative, derivative practice, (f'(x))", "Meta Description: Learn how to find the derivative of (f(x) = 3x^3 - 5x^2 + 6x - 4) step-by-step. Includes power rule application, differentiation of each term, and final result with explanation. Ideal for calculus students and educators."]