It makes the left-hand side look like the result of a product-rule differentiation. Just in case a picture helps... suppose
is the product rule - straight continuous lines differentiating downwards (integrating up) with respect to x. Then we're trying to fit the left-hand side...
... along the bottom row. As it comes, it won't fit (and satisfy the rule). But if we accomodate the chain rule...
... inside the (legs-uncrossed version of the) product rule...
... we can see how to fix it...
Hope that helps.
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).
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Don't integrate - balloontegrate!
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