I would say it is not the Fundamental Theorem of Calculus, but rather some notion connecting limits and continuity, perhaps the $(\epsilon,\delta)$-definitions of limits and continuity. But I would be interested to learn from experienced calculus instructors, as it would help me in courses that have Calculus as a prerequisite, to understand where to anticipate weaknesses.
For the usual meaning of "Calculus 1" (or "Intro to Calculus"), see Is there a more telling name for “Calculus 2”? See also Wikipedia's List of calculus topics.
Summary by 16Aug2019: I did not anticipate such lack of consensus.
- @vonbrand: "the whole $\epsilon/\delta$ dance."
- @SueVanHattum: implicit derivatives and optimization
- @guest2: "multistep problem solving"
- @Maesumi: "the concept of a variable and that of a function"
- @JamesSCook: "the chain rule is most troublesome"
- @user1527: "the point of the mean value theorem"