Find the derivative of the function f(x) = 3x³ - 5x² + 2x - 7.
["# Find the Derivative of the Function f(x) = 3x³ - 5x² + 2x - 7", "Understanding derivatives is a fundamental concept in calculus, especially when analyzing the behavior of functions. If you’re asking how to find the derivative of the function\n[ f(x) = 3x^3 - 5x^2 + 2x - 7, ]\nyou're on the right track. Derivatives help us determine slopes, rates of change, and critical points—key insights for both pure math and applied fields like physics and engineering.", "## What Is a Derivative?", "In simple terms, the derivative of a function at a point gives the instantaneous rate of change of the function at that point. It’s the slope of the tangent line to the function’s graph at a specific (x)-value.", "For polynomial functions, computing the derivative is straightforward using basic rules: the power rule, constant multiple rule, and sum rule.", "---", "## Applying the Power Rule to f(x)", "The function is:\n[ f(x) = 3x^3 - 5x^2 + 2x - 7 ]", "We apply the power rule:\nIf ( f(x) = ax^n ), then ( f'(x) = a \cdot n \cdot x^{n-1} ).", "Let’s compute each term individually.", "### Derivative of ( 3x^3 )", "[\n\frac{d}{dx}(3x^3) = 3 \cdot 3x^{3-1} = 9x^2\n]", "### Derivative of ( -5x^2 )", "[\n\frac{d}{dx}(-5x^2) = -5 \cdot 2x^{2-1} = -10x\n]", "### Derivative of ( 2x )", "The derivative of ( x ) is 1, so:\n[\n\frac{d}{dx}(2x) = 2 \cdot 1 = 2\n]", "### Derivative of ( -7 ) (constant)", "The derivative of any constant is zero:\n[\n\frac{d}{dx}(-7) = 0\n]", "---", "## Combining All Derivatives", "Now, add up the derivatives of each term:", "[\nf'(x) = 9x^2 - 10x + 2 + 0 = 9x^2 - 10x + 2\n]", "---", "## Final Answer", "[\n\boxed{f'(x) = 9x^2 - 10x + 2}\n]", "---", "## Why This Matters", "Knowing the derivative helps identify:\n- Critical points where ( f'(x) = 0 ), useful for finding maxima and minima\n- Increasing/Decreasing intervals using the sign of the derivative\n- Tangent slopes at any given (x), aiding in curve sketching", "Whether you’re learning calculus, preparing for exams, or solving real-world optimization problems, mastering derivative rules is essential.", "---", "## Quick Summary – Rules Recap", "| Function Type | Derivative Rule | Example |\n|---------------------|---------------------------------|------------------|\n| ( ax^n ) | ( a \cdot n x^{n-1} ) | ( 3x^3 \ o 9x^2 )|\n| Constant | ( \frac{d}{dx}(c) = 0 ) | ( -7 \ o 0 ) |\n| ( x \ o 1 \cdot x ) | ( \frac{d}{dx}(x) = 1 ) | ( 2x \ o 2 ) |", "---", "Key Takeaway: The derivative of ( f(x) = 3x^3 - 5x^2 + 2x - 7 ) is ( f'(x) = 9x^2 - 10x + 2 ), obtained by applying the power rule term-by-term.", "If you enjoyed this breakdown, explore related topics like implicit differentiation or applications of derivatives in physics and economics!"]
