Just in case a picture helps...
... 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).
Sorry you never got a picture of the cup itself, which is what you were probably hoping for! But the most important thing with related rates is usually the chain rule.
Don't integrate - balloontegrate!