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TBF "what calculus is" -- i.e., the level of rigor at which topics are taught -- varies a lot even among college courses called "calculus"...

In particular, IMO the distance between AP calculus and honors calculus at an ivy is larger than the difference between this material and AP calculus.

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Up to a point, but does it not all grow out of the concept of limits?

I would say that the fundamental idea to understand is the concept of changing ratios.

Limits seems to me to be more a tool in order to calculate the ratios than a fundamental concept.

I agree with that. You can hand limits as a tool to someone already comfortable with the concepts and present them as a way to quantify their intuitive understanding.

But without limits, you cannot actually do much with the concepts. Whether you call limits a concept or a tool, they are essential to the practice of calculus, and, historically, to its emergence (and not trivially so, either.)

Yeah but you can do well on the AP calculus exam without understanding limits, if you can calculate derivatives and solve some word problems related to derivatives.

Hence, the gap between "5 year old calculus" and "high school calculus" is smaller than the gap between "high school calculus" and "actual calculus"

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