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Volume I · MMXXVI AP Calculus AB/BC
Library› AP Calculus AB/BC› Unit 6› The Fundamental Theorem of Calculus, Part 1
⁂   AP Calculus AB/BC · Unit 6

2. The Fundamental Theorem of Calculus, Part 1

Topics: ftc, ftc-part1, accumulation-functions, chain-rule-with-variable-bounds.

Difficulty 3/5 Review interval 4 days

**FTC Part 1** connects differentiation and integration by telling us how to differentiate an accumulation function. If $g(x) = \int_a^x f(t)\,dt$ where $f$ is continuous, then $g'(x) = f(x)$. Intuitively: the rate of change of the accumulated area equals the height of the curve at the current point. **When the upper bound is a function of $x$** (composite case), apply the chain rule: $$\frac{d}{dx}\int_a^{u(x)} f(t)\,dt = f(u(x))\cdot u'(x)$$ This composite form is the version most commonly tested on the AP exam.

Key Formulas

<span class="math-block">\[\frac{d}{dx}\int_a^{x} f(t)\,dt = f(x)\]</span>
<span class="math-block">\[\frac{d}{dx}\int_a^{u(x)} f(t)\,dt = f(u(x))\cdot u'(x)\]</span>

Common Errors

Watch out for
  • Forgetting the chain-rule factor u'(x) when the upper bound is a function of x — the most cited error on FTC Part 1 problems.
  • Differentiating the integrand f(t) instead of evaluating it at the upper bound.
  • Confusing FTC Part 1 (differentiation of integral) with FTC Part 2 (evaluation of integral by antiderivative).

Practice Questions

Practice MCQs for this lesson ›

Originality note

Our worked solutions and practice questions are original instructional content created by Tian2 AP. They are aligned to the concepts and skills described in College Board’s Course and Exam Description and are not reproductions of, or affiliated with, College Board’s official materials.