Just in case a picture helps...
... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to x, and the straight dashed line similarly but with respect to the dashed balloon expression (the inner function of the composite which is subject to the chain rule).
OK, but why does
This time we can make an 'internal' substitution, replacing x with sec \theta...
As before, it's basically a clockwise journey...
Hope this helps.
Alternatively, and probably much better, combine both in one... (spoiler to accommodate width)
Don't integrate - balloontegrate!
Balloon Calculus: Gallery
Balloon Calculus Drawing with LaTeX and Asymptote!