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

["# Find the Derivative of the Function ( f(x) = 3x^3 - 5x^2 + 2x - 7 )", "Understanding derivatives is a fundamental part of calculus, especially in fields like physics, engineering, and economics. If you're studying functions and want to find the derivative of ( f(x) = 3x^3 - 5x^2 + 2x - 7 ), this article guides you step-by-step through the process.", "## What is a Derivative?", "The derivative of a function represents its rate of change at any given point. For polynomials like ( f(x) = 3x^3 - 5x^2 + 2x - 7 ), finding the derivative means applying basic differentiation rules to each term.", "## Step-by-Step: How to Differentiate ( f(x) )", "### Step 1: Recall the basic derivative rules\n- The derivative of ( x^n ) is ( nx^{n-1} ) (power rule).\n- The derivative of a constant is 0.\n- Constants multiplied by functions can be treated like the constant factor (constant multiple rule).", "### Step 2: Differentiate each term", "Break down the function term-by-term:", "1. ( 3x^3 )\n Apply the power rule:\n [\n \frac{d}{dx}(3x^3) = 3 \cdot 3x^{3-1} = 9x^2\n ]", "2. ( -5x^2 )\n Apply the power rule:\n [\n \frac{d}{dx}(-5x^2) = -5 \cdot 2x^{2-1} = -10x\n ]", "3. ( 2x )\n Write ( 2x ) as ( 2x^1 ):\n [\n \frac{d}{dx}(2x) = 2 \cdot 1x^{1-1} = 2\n ]", "4. ( -7 )\n This is a constant, so its derivative is 0:\n [\n \frac{d}{dx}(-7) = 0\n ]", "### Step 3: Combine the derivatives", "Now add the derivatives of each term:\n[\nf'(x) = 9x^2 - 10x + 2 + 0\n]", "### Final Answer:\n[\n\boxed{f'(x) = 9x^2 - 10x + 2}\n]", "---", "## Why Learning Derivatives Matters", "Knowing how to differentiate functions like ( f(x) = 3x^3 - 5x^2 + 2x - 7 ) enables you to determine slopes of curves, locate critical points (maxima, minima), and analyze behavior of real-world models. Whether you're optimizing profit functions or modeling motion, derivatives are essential tools in calculus.", "Ready to tackle more calculus? Practice by differentiating other polynomials and explore applications like curve sketching and optimization.", "---", "Keywords: find the derivative of ( f(x) = 3x^3 - 5x^2 + 2x - 7 ), derivative rules, calculus tutorial, polynomial differentiation, derivative of cubic function.\nMeta Description: Learn how to find the derivative of ( f(x) = 3x^3 - 5x^2 + 2x - 7 ) step-by-step with rules, examples, and applications. Perfect for students of calculus and applied math."]