What is the derivative of \( f(x) = 3x^3 - 5x^2 + 2x - 7 \) at \( x = 1 \)?

["# Understanding the Derivative of ( f(x) = 3x^3 - 5x^2 + 2x - 7 ) at ( x = 1 )", "When studying calculus, one of the fundamental concepts you encounter is the derivative—a powerful tool that helps us understand how functions change at any given point. If you're wondering what the derivative of the function ( f(x) = 3x^3 - 5x^2 + 2x - 7 ) is at ( x = 1 ), you're engaging with one of the most essential applications of differentiation.", "## What is a Derivative?", "The derivative of a function at a specific point represents the instantaneous rate of change of that function at that point. Geometrically, it corresponds to the slope of the tangent line to the graph of the function at ( x = 1 ).", "## Step-by-Step: Finding ( f'(x) )", "To find the derivative of ( f(x) = 3x^3 - 5x^2 + 2x - 7 ), we apply basic differentiation rules term by term.", "- The derivative of ( 3x^3 ) is ( 9x^2 ) (using ( \frac{d}{dx} [x^n] = nx^{n-1} ))\n- The derivative of ( -5x^2 ) is ( -10x )\n- The derivative of ( 2x ) is ( 2 )\n- The derivative of the constant ( -7 ) is ( 0 )", "Putting it all together, the derivative is:\n[\nf'(x) = 9x^2 - 10x + 2\n]", "## Evaluating the Derivative at ( x = 1 )", "Now that we have the derivative, we substitute ( x = 1 ) into the expression:\n[\nf'(1) = 9(1)^2 - 10(1) + 2 = 9 - 10 + 2 = 1\n]", "## Why ( f'(1) = 1 ) Matters", "This result means that at ( x = 1 ), the function ( f(x) ) is increasing at a rate of 1 unit per unit increase in ( x ). The slope of the tangent line at this point is flat but positive—indicating a weak but upward trend.", "In practical terms, if ( f(x) ) represents any real-world quantity—like position, cost, or temperature—then ( f'(1) = 1 ) tells us how quickly that quantity is changing precisely at that moment.", "## Conclusion", "Calculating the derivative of ( f(x) = 3x^3 - 5x^2 + 2x - 7 ) gives ( f'(x) = 9x^2 - 10x + 2 ). Evaluating this at ( x = 1 ) yields ( f'(1) = 1 ). This derivative value is not just a number—it’s a snapshot of the function’s instantaneous behavior, vital for optimization, motion analysis, and modeling dynamic systems.", "So next time you see ( f'(1) ), remember: you’re not just solving an algebra problem—you’re unlocking key insights into how functions behave and evolve.", "---", "Keywords: derivative of ( f(x) = 3x^3 - 5x^2 + 2x - 7 ), ( f'(1) ), calculus, find derivative, instantaneous rate of change, mathematical derivative, teaching derivatives, derivative rules, output ( f'(x) = 9x^2 - 10x + 2 )", "Meta Description: Learn how to compute the derivative of ( f(x) = 3x^3 - 5x^2 + 2x - 7 ) at ( x = 1 ), why it equals 1, and what this means in calculus and real-world applications."]