Just in case a picture helps...

... where (key in spoiler) ...

Spoiler:

You can simplify the picture by losing many of the labels. Indeed you might have just values in most of the balloons.

Then do similar for dt, where the continuous lines mean with respect to t, and s is held constant.

Then do similar for the second derivatives.

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Don't integrate - balloontegrate!

Balloon Calculus; standard integrals, derivatives and methods

Balloon Calculus Drawing with LaTeX and Asymptote!