Evaluating at \(x = 2\): \(f'(2) = 6(2) - 2 = 12 - 2 = 10\).

["Evaluating ( f'(2) = 6x - 2 ) at ( x = 2 ): A Step-by-Step Explanation", "Understanding how to evaluate the derivative of a function at a specific point is essential in calculus, especially when analyzing the behavior of functions in science, economics, and engineering. In this article, we evaluate ( f'(2) ) for the derivative function ( f'(x) = 6x - 2 ), explaining each step clearly and why the final result is ( f'(2) = 10 ).", "---", "### What Does ( f'(2) ) Mean?", "The notation ( f'(2) ) represents the instantaneous rate of change of the original function ( f(x) ) at ( x = 2 ). But sometimes, instead of finding the derivative function ( f'(x) ) first, it’s possible to directly evaluate the derivative at a point using rules like the power rule, constant multiple rule, and constant function rule.", "Here, we are given the derivative as:", "[\nf'(x) = 6x - 2\n]", "This is a linear function, and evaluating it at ( x = 2 ) means substituting ( x = 2 ) into the expression.", "---", "### Step-by-Step Evaluation at ( x = 2 )", "Start with the derivative expression:", "[\nf'(x) = 6x - 2\n]", "Substitute ( x = 2 ):", "[\nf'(2) = 6(2) - 2\n]", "Multiply first:", "[\nf'(2) = 12 - 2\n]", "Subtract:", "[\nf'(2) = 10\n]", "---", "### Why This Makes Sense", "The derivative ( f'(x) = 6x - 2 ) describes a straight line with slope 6 and y-intercept ( -2 ). At any point, including ( x = 2 ), the slope — and thus the rate of change of the function — is simply the value of the derivative at that point.", "So evaluating ( f'(2) ) is equivalent to asking: "What is the slope of ( f(x) ) at ( x = 2 )?" The algebraic computation confirms it’s 10.", "---", "### Alternative: Using Definition of Derivative", "While not necessary here due to the known ( f'(x) ), we can briefly recall the limit definition:", "[\nf'(2) = \lim_{h \ o 0} \frac{f(2+h) - f(2)}{h}\n]", "The function ( f(x) ) obtained by integrating ( f'(x) = 6x - 2 ) is:", "[\nf(x) = \int (6x - 2) , dx = 3x^2 - 2x + C\n]", "Then:", "[\nf'(2) = \ ext{slope at } x=2 = 6(2) - 2 = 10\n]", "Again, confirming the same result.", "---", "### Practical Significance", "Knowing ( f'(2) = 10 ) tells us:", "- The function is increasing steeply at ( x = 2 ), since the derivative is positive and large.\n- The rate at which ( f(x) ) changes when ( x = 2 ) is 10 units per unit increase in ( x ).", "This insight is vital in applications such as modeling growth, optimizing costs, or predicting physical systems.", "---", "### Conclusion", "Evaluating ( f'(2) ) for ( f'(x) = 6x - 2 ) is straightforward: substitute ( x = 2 ), multiply by 6, subtract 2, and compute. The result, ( f'(2) = 10 ), gives the exact rate of change of ( f(x) ) at that point. Mastering such evaluations strengthens your calculus foundation and supports advanced problem-solving.", "---", "Keywords: evaluate ( f'(2) ), calculate derivative at 2, mathematical evaluation, derivative rules, instantaneous rate of change, ( f'(2) = 6(2) - 2 ), calculus example."]