Find the derivative of \(f(x) = 4x^3 - 5x^2 + 2x - 7\).

["# Find the Derivative of ( f(x) = 4x^3 - 5x^2 + 2x - 7 )", "Understanding derivatives is crucial in calculus, especially when analyzing the rate of change of functions in fields like physics, engineering, economics, and data analysis. In this article, we’ll walk through an easy step-by-step process to find the derivative of the function:", "[\nf(x) = 4x^3 - 5x^2 + 2x - 7\n]", "## What is a Derivative?", "The derivative of a function at any point ( x ) represents the slope of the tangent line to the function at that point. It tells us how the function ( f(x) ) changes as ( x ) changes—an essential concept in optimization, motion analysis, and more.", "## Rules of Differentiation", "To find the derivative, we apply standard differentiation rules:", "- Power Rule:\n If ( f(x) = ax^n ), then ( f'(x) = a \cdot n x^{n-1} ).\n- Constant Rule:\n The derivative of a constant is 0.\n- Constant Multiple Rule:\n The derivative of ( a \cdot g(x) ) is ( a \cdot g'(x) ).\n- Sum/Difference Rule:\n The derivative of a sum or difference is the sum or difference of the derivatives.", "## Step-by-Step Derivative Calculation", "### Step 1: Differentiate each term individually", "The function is a polynomial:\n[\nf(x) = 4x^3 - 5x^2 + 2x - 7\n]", "Differentiate term by term:", "1. First term: ( 4x^3 )\n Apply the power rule:\n [\n \frac{d}{dx}(4x^3) = 4 \cdot 3x^{2} = 12x^2\n ]", "2. Second term: ( -5x^2 )\n Apply the power rule:\n [\n \frac{d}{dx}(-5x^2) = -5 \cdot 2x^{1} = -10x\n ]", "3. Third term: ( 2x )\n Since ( x = x^1 ):\n [\n \frac{d}{dx}(2x) = 2 \cdot 1x^{0} = 2\n ]", "4. Fourth term: ( -7 ) (a constant)\n The derivative of any constant is zero:\n [\n \frac{d}{dx}(-7) = 0\n ]", "### Step 2: Add the derivatives together", "Summing all the derivatives term by term gives:", "[\nf'(x) = 12x^2 - 10x + 2 + 0\n]", "### Final Answer:", "[\n\boxed{f'(x) = 12x^2 - 10x + 2}\n]", "## Why This Matters", "The derivative ( f'(x) = 12x^2 - 10x + 2 ) tells you how rapidly the function ( f(x) ) grows or shrinks at any point ( x ). For example:", "- Positive values of ( f'(x) ) indicate the function is increasing.\n- Negative values show the function is decreasing.\n- The expression helps locate critical points where the slope is zero, useful for finding local maxima and minima.", "## Summary", "Finding the derivative of ( f(x) = 4x^3 - 5x^2 + 2x - 7 ) involves:", "1. Applying the power rule term by term.\n2. Using constant rules.\n3. Combining results with the sum rule.", "This process forms the foundation for advanced calculus applications, including optimization and modeling real-world phenomena.", "If you're studying calculus or working with functions, mastering derivatives through practice is key—this function serves as a perfect starting point. For more practice, try derivatives of higher-degree polynomials or functions with mixed terms!", "---", "Keywords: find the derivative, derivative of polynomial, power rule, calculus tutorial, algebra help, rate of change, calculus rules, ( f(x) = 4x^3 - 5x^2 + 2x - 7 ), derivative step-by-step."]