http://www.ballooncalculus.org/asy/chain.png
... is the chain rule. Straight continuous lines differentiate downwards (integrate up) with respect to the main variable (in this case t), 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).

For the subsequent part:

http://www.ballooncalculus.org/asy/prod.png
... is the product rule, where, again, straight continuous lines are differentiating downwards with respect to x.