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

["# Find the Derivative of ( f(x) = x^3 - 3x^2 + 5x - 7 ) at ( x = 2 )", "Calculating derivatives is a fundamental skill in calculus, essential for understanding function behavior, optimization, and rates of change. In this article, we’ll walk through how to find the derivative of ( f(x) = x^3 - 3x^2 + 5x - 7 ) and evaluate it at ( x = 2 ).", "---", "## Step 1: Understand What a Derivative Represents", "The derivative of a function at a point gives the instantaneous rate of change of the function at that point — essentially, the slope of the tangent line.", "---", "## Step 2: Differentiate the Function", "Given:\n[\nf(x) = x^3 - 3x^2 + 5x - 7\n]", "Apply standard differentiation rules term by term:", "- The derivative of ( x^3 ) is ( 3x^2 )\n- The derivative of ( -3x^2 ) is ( -6x )\n- The derivative of ( 5x ) is ( 5 )\n- The derivative of constant ( -7 ) is ( 0 )", "Putting it all together:\n[\nf'(x) = 3x^2 - 6x + 5\n]", "---", "## Step 3: Evaluate the Derivative at ( x = 2 )", "Now substitute ( x = 2 ) into ( f'(x) ):\n[\nf'(2) = 3(2)^2 - 6(2) + 5\n]\n[\n= 3(4) - 12 + 5\n]\n[\n= 12 - 12 + 5 = 5\n]", "---", "## Final Answer", "[\n\boxed{f'(2) = 5}\n]", "---", "## Why This Matters", "Understanding that ( f'(2) = 5 ) means the function is increasing at a slope of 5 at ( x = 2 ). This process shows how symbolic differentiation enables precise, mathematical insights into function behavior — invaluable in physics, economics, engineering, and beyond.", "---", "Keywords: derivative, find derivative, derivative of ( x^3 - 3x^2 + 5x - 7 ), calculus, ( f'(2) ), differentiation rules, function analysis.\nMeta Description: Learn how to compute the derivative of ( f(x) = x^3 - 3x^2 + 5x - 7 ) and evaluate it at ( x = 2 ) with step-by-step explanation.", "---", "Need more calculus help? Explore related articles on chain rule, product rule, or apply derivatives to solve real-world optimization problems."]