... 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.
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 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.