Calculate the derivative of \( g(x) = 5x^3 - 3x^2 +

["# Calculate the Derivative of ( g(x) = 5x^3 - 3x^2 + 2x - 7 )", "Understanding derivatives is essential in calculus, especially for studying functions in mathematics, engineering, physics, and economics. In this article, we’ll walk through how to calculate the derivative step by step for the function:", "[\ng(x) = 5x^3 - 3x^2 + 2x - 7\n]", "We’ll use the basic derivative rules to find the slope of the function at any point ( x ), helping you apply these principles to more complex functions.", "---", "## What Is a Derivative?", "The derivative of a function ( g(x) ) at a point ( x ), denoted ( g'(x) ), represents the rate of change of ( g(x) ) with respect to ( x ). Geometrically, it gives the slope of the tangent line to the curve of ( g(x) ) at ( x ).", "---", "## Applying the Power Rule", "The primary tool in differentiating polynomial functions is the power rule, which states:", "> If ( f(x) = ax^n ), then ( f'(x) = a \cdot n \cdot x^{n-1} )", "This rule applies to each term in a polynomial separately.", "---", "## Step-by-Step Derivative Calculation", "Let’s differentiate each term of ( g(x) = 5x^3 - 3x^2 + 2x - 7 ):", "### 1. Differentiate ( 5x^3 )", "- Apply the power rule: coefficient ( 5 ), exponent ( 3 )\n- Multiply: ( 5 \cdot 3 = 15 )\n- Reduce exponent: ( x^{3-1} = x^2 )\n- Result: ( \frac{d}{dx}(5x^3) = 15x^2 )", "### 2. Differentiate ( -3x^2 )", "- Apply the power rule: coefficient ( -3 ), exponent ( 2 )\n- Multiply: ( -3 \cdot 2 = -6 )\n- Exponent becomes ( 1 ): ( x^1 = x )\n- Result: ( \frac{d}{dx}(-3x^2) = -6x )", "### 3. Differentiate ( 2x )", "- Rewrite ( 2x ) as ( 2x^1 )\n- Apply power rule: ( 2 \cdot 1 = 2 ), exponent becomes ( 0 ): ( x^0 = 1 )\n- Result: ( \frac{d}{dx}(2x) = 2 \cdot 1 = 2 )", "### 4. Differentiate the constant ( -7 )", "- The derivative of any constant is 0\n- Result: ( \frac{d}{dx}(-7) = 0 )", "---", "## Combine All Terms", "Now add the derivatives of each term:", "[\ng'(x) = 15x^2 - 6x + 2 + 0\n]", "---", "## Final Answer", "[\n\boxed{g'(x) = 15x^2 - 6x + 2}\n]", "---", "## Why Learn Derivatives?", "Computing derivatives equips you with powerful tools to:\n- Determine local maxima and minima\n- Analyze function growth or decay\n- Solve optimization problems\n- Model real-world dynamics in physics, economics, and beyond", "---", "## Summary", "To summarize, differentiating the function:\n[\ng(x) = 5x^3 - 3x^2 + 2x - 7\n]\nyields the derivative:\n[\ng'(x) = 15x^2 - 6x + 2\n]", "Use the power rule systematically to break down each term, combine like terms, and obtain your result efficiently. Clear understanding of derivatives is key to mastering calculus.", "---", "If you want to practice more, try differentiating other polynomial functions using the same step-by-step method! Whether you're in high school, college, or self-studying, mastering derivatives opens doors to advanced mathematical concepts."]