That's a nice clip! Perhaps anyone who is helped by it (especially the bit where we circle the function to help emphasise inner or outer) will also be interested in this kind of thing...
... where straight continuous lines differentiate (downwards) with respect to x, and the dashed line with respect to the dashed balloon expression - so that the triangular network satisfies the rule.
E.g.,
We can use the same approach to integrate - just building up the picture from the bottom end instead of the top. (See Balloon Calculus: worked examples from past papers and my other posts to see the process.) E.g.,
"Don't integrate - balloontegrate!"
I don't think so.
I don't think the terms, as applied to calculus, are analogous to their normal meanings.
It's simply defined that:
A derivative is the rate of change of a function.
The process of finding a derivative is differentiation.
The result of n differentiations is the nth derivative, or nth differential of a function.