Just in case a picture helps...

... where

... 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).

Hope that might help - follow round clockwise from bottom left. Full size pic in here:

Spoiler:

By the way, you're aware of Pythagoras' involvement for simplifying to ?

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Don't integrate - balloontegrate!

Balloon Calculus: Gallery

Balloon Calculus Drawing with LaTeX and Asymptote!