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 subject to the chain rule).
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Don't integrate - balloontegrate!
Balloon Calculus: Gallery
Balloon Calculus Drawing with LaTeX and Asymptote!
Yes, and it won't create that 'powers' problem, will it?
But now the goal posts have moved, let's not bicker... I may have provoked you in the first place by daring to introduce my pic with 'quicker by balloon!', which I edited out within the minute because I remembered that this is only true for quotients already having a power in the denominator - see http://www.ballooncalculus.org/examp...ence.html#quot if interested - hope so!
Cheers
Tom