Calculate the derivative of the function \(f(x) = 4x^3 - 5x^2 + 2x - 7\).
["# Calculate the Derivative of the Function ( f(x) = 4x^3 - 5x^2 + 2x - 7 )", "Understanding calculus is essential in fields like engineering, physics, economics, and data science. One of the most fundamental concepts is the derivative, which provides the rate of change of a function at any point. In this article, we’ll walk through step-by-step how to compute the derivative of the cubic polynomial function:", "[\nf(x) = 4x^3 - 5x^2 + 2x - 7\n]", "## What is a Derivative?", "The derivative, denoted ( f'(x) ), measures how a function ( f(x) ) changes as its input ( x ) changes. For polynomials, the derivative can be calculated using basic rules:", "- Power Rule: If ( f(x) = ax^n ), then ( f'(x) = a \cdot n \cdot x^{n-1} )\n- Constant Rule: The derivative of a constant is zero.\n- Linearity: The derivative of a sum/difference follows term-by-term differentiation.", "---", "## Step-by-Step Derivative Calculation", "Let’s compute ( f'(x) ) for ( f(x) = 4x^3 - 5x^2 + 2x - 7 ) by differentiating each term individually.", "### 1. Differentiate ( 4x^3 )", "Using the power rule:", "[\n\frac{d}{dx}(4x^3) = 4 \cdot 3 \cdot x^{3-1} = 12x^2\n]", "### 2. Differentiate ( -5x^2 )", "Again applying the power rule:", "[\n\frac{d}{dx}(-5x^2) = -5 \cdot 2 \cdot x^{2-1} = -10x\n]", "### 3. Differentiate ( 2x )", "Applying the power rule with ( n = 1 ):", "[\n\frac{d}{dx}(2x) = 2 \cdot 1 \cdot x^{1-1} = 2 \cdot 1 = 2\n]", "### 4. Differentiate the constant ( -7 )", "Constants vanish under differentiation:", "[\n\frac{d}{dx}(-7) = 0\n]", "---", "## Combine the results", "Adding all derivatives together gives the full derivative of ( f(x) ):", "[\nf'(x) = 12x^2 - 10x + 2 + 0\n]", "Thus,", "[\n\boxed{f'(x) = 12x^2 - 10x + 2}\n]", "---", "## Why Is This Derivative Important?", "- Slope Interpretation: The derivative at any point gives the slope of the tangent line to the curve ( y = f(x) ).\n- Critical Points: Setting ( f'(x) = 0 ) helps locate maxima, minima, and inflection points.\n- Application in Optimization: Derivatives are critical in maximizing profits, minimizing costs, and analyzing rates of change in real-world scenarios.", "---", "## Summary", "Calculating the derivative of ( f(x) = 4x^3 - 5x^2 + 2x - 7 ) is straightforward using basic differentiation rules. By applying the power rule term-by-term, we efficiently found:", "[\n\boxed{f'(x) = 12x^2 - 10x + 2}\n]", "Mastering derivatives empowers deeper insights into function behavior and enables solving complex problems across science and engineering disciplines.", "---", "## Further Reading and Resources", "- Khan Academy: Calculus Derivatives\n- Paul’s Online Math Notes: Derivatives\n- Desmos Calculus Visualization: Explore how derivatives relate to graphs dynamically.", "Enable your learning journey — understanding derivatives opens doors to advanced calculus concepts and practical applications!"]
