Find the derivative of \( f(x) = 3x^3 - 5x^2 + 2x - 7 \).

["# Find the Derivative of ( f(x) = 3x^3 - 5x^2 + 2x - 7 )", "Understanding derivatives is a fundamental aspect of calculus, especially when analyzing rates of change and optimizing functions. If you’re studying algebra or pre-calculus, one of the first derivative problems you’ll encounter is differentiating polynomial functions. In this article, we’ll clearly walk through how to find the derivative of\n[ f(x) = 3x^3 - 5x^2 + 2x - 7 ],\nusing the basic rules of differentiation.", "## What is a Derivative?", "The derivative of a function at a point gives the slope of the tangent line to the function’s graph at that point. For polynomial functions, differentiation follows a few clear rules that make the process systematic and efficient.", "---", "## Step-by-Step Derivative Calculation", "Let’s compute ( f'(x) ), the derivative of ( f(x) = 3x^3 - 5x^2 + 2x - 7 ).", "We apply the power rule of differentiation, which states:\n[\n\frac{d}{dx}[x^n] = nx^{n-1}\n]\nWe also use:\n- The derivative of a constant is zero.\n- Derivatives are linear, so we can differentiate term by term.", "### Differentiate each term:", "1. First term: ( 3x^3 )\n [\n \frac{d}{dx}[3x^3] = 3 \cdot 3x^{3-1} = 9x^2\n ]", "2. Second term: ( -5x^2 )\n [\n \frac{d}{dx}[-5x^2] = -5 \cdot 2x^{2-1} = -10x\n ]", "3. Third term: ( 2x )\n [\n \frac{d}{dx}[2x] = 2 \cdot 1x^{1-1} = 2\n ]", "4. Fourth term: ( -7 )\n [\n \frac{d}{dx}[-7] = 0 \quad \ ext{(constant)}\n ]", "---", "### Combine the derivatives:", "Adding all the derivatives of each term together, we get:\n[\nf'(x) = 9x^2 - 10x + 2\n]", "---", "## Final Answer", "[\n\boxed{f'(x) = 9x^2 - 10x + 2}\n]", "---", "## Why This Matters", "Knowing how to differentiate functions like ( f(x) = 3x^3 - 5x^2 + 2x - 7 ) helps you understand key concepts in science, economics, engineering, and machine learning. Whether you’re finding optimization points, modeling motion, or analyzing growth rates, derivatives are powerful tools.", "If you found this explanation helpful, share it with fellow learners or explore related topics like the chain rule, product rule, and higher-order derivatives to deepen your calculus mastery.", "---", "Keywords: derivative, find derivative, ( f(x) = 3x^3 - 5x^2 + 2x - 7 ), differentiation rules, power rule, calculus tutorial, find f’(x), algebra, continuous learning, math help."]