At \(x = 1\): \(f'(1) = rac{2(1)}{1 + 1} = rac{2}{2} = 1\).

["Understanding the Derivative at ( x = 1 ): A Clear Step-by-Step Explanation", "When analyzing functions in calculus, one of the most important concepts is the derivative — a measure of how a function changes at any given point. For many students, understanding how to compute the derivative at a specific value like ( x = 1 ) can be confusing, but breaking it down step by step makes it easier and more intuitive.", "Evaluating the derivative at ( x = 1 ):\nSuppose we are given the function ( f(x) ) such that at ( x = 1 ), the derivative is computed using the formula:", "[\nf'(1) = \frac{2(1)}{1 + 1}\n]", "This expression derives from applying the standard derivative rule (such as quotient rule or function composition) to the original function definition — specifically, the numerator’s coefficient (here 2) is divided by the denominator (here ( 1 + x )) evaluated at ( x = 1 ).", "Let’s walk through the calculation step by step:", "1. Identify the numerator coefficient:\n The top of the fraction is ( 2(1) = 2 ), representing a constant multiplier applied to the simplified function behavior at that point.", "2. Evaluate the denominator at ( x = 1 ):\n The denominator is ( 1 + 1 = 2 ), representing the rate denominator from the function’s internal structure.", "3. Compute the derivative:\n [\n f'(1) = \frac{2}{2} = 1\n ]", "This result tells us that at ( x = 1 ), the instantaneous rate of change — or slope of the tangent line — of the function ( f(x) ) is exactly 1. Geometrically, this means that if you draw the function near ( x = 1 ), the curve rises one unit vertically for every one unit horizontally — consistent with a slope of 1.", "Why this matters:\nUnderstanding derivatives at specific points is crucial in many applications, from optimizing functions in engineering to modeling motion in physics. Computing ( f'(1) ) directly helps uncover local function behavior quickly without needing to graph the entire curve.", "In summary, the equation\n[\nf'(1) = \frac{2(1)}{1 + 1} = \frac{2}{2} = 1\n]\nis a clear and correct evaluation of the derivative at ( x = 1 ), rooted in standard calculus rules and interpretation.", "---", "Keywords: derivative at ( x = 1 ), ( f'(1) ), how to compute derivatives, calculus step-by-step, instantaneous rate of change, derivative evaluation, math tutorial.", "Meta Title: Understanding ( f'(1) = \frac{2}{2} = 1 ): Derivative at ( x = 1 ) Explained Simply\nMeta Description: Learn how to compute ( f'(1) ) using the derivative definition—step-by-step. Discover why ( f'(1) = 1 ) when ( f'(1) = \frac{2(1)}{1 + 1} ). Clear math explanation for students and beginners."]