Are you sure? Here's a pic - I'll edit details and quite possibly corrections...

Actually, the chain rule here is kind of trivial, so we could just use the product rule alone...

... where...

... is the product rule. Straight continuous lines differentiate downwards (integrate up) with respect to x, and then, because of the product rule, the whole of the bottom is the derivative of the whole of the top.

Notice I've tweaked the second balloon along the bottom, just for a common denominator.

OK - where does that get us in view of your proposed answer? Have you checked question and answer? Maybe its me...

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Don't integrate - balloontegrate!

http://www.ballooncalculus.org/examples/gallery.html

http://www.ballooncalculus.org/asy/doc.html