Fundamental Theorem Of Calculus

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If a function F is defined as the integral from one to x of the reciprocal of t, then differentiate F and identify the value of its derivative at the point x equals two.

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This problem directly invokes the first Fundamental Theorem of Calculus, the result that ties accumulation to instantaneous rate of change and forms the conceptual core of the entire subject. The theorem states that if F(x) = \int_a^x f(t)\,dt with f continuous on the interval, then F'(x) = f(x). The deep idea is that differentiating an accumulation function simply returns the integrand evaluated at the moving upper limit, so no explicit antiderivative is needed. Here f(t) = 1/t is continuous for t > 0, hence F'(x) = 1/x. Substituting x = 2 gives F'(2) = 1/2. As a reassuring cross-check, F(x) itself equals \ln x, whose derivative is indeed 1/x, fully consistent with the theorem. Option 2 inverts the reciprocal incorrectly. Option \ln 2 is the value of F(2) itself, not its derivative, confusing the function with its slope. Option 1 would require evaluating at the lower limit x = 1 rather than at x = 2. As a final plausibility check, since 1/t is positive and decreasing, F should be increasing with a slope that shrinks as x grows, and the value 1/2 at x = 2 sits exactly within that decreasing-slope pattern.

Fundamental Theorem Of Calculus

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