Just in case a picture helps...

... where

... is the chain rule, and...

... the product 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 subject to the chain rule).

So what you've got here is two chains inside a product-rule (the legs-uncrossed version). That's just my preference - you could do it as a product (or even a quotient - yeughh!) inside a chain.

Anyway, if you feel up to filling in the blanks and simplifying, notice I've tweaked the third balloon along for the sake of the common denominator.

Hope that helps, or doesn't further confuse.

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Don't integrate - balloontegrate!

Balloon Calculus: Gallery

Balloon Calculus Drawing with LaTeX and Asymptote!