At 8:20 in Lecture 4 of course 18-01sc, the professor writes "u * delta v / delta x". Shouldn't that be "u(x) * delta v / delta x"? I don't understand why the (x) is dropped.

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Just being a bit sloppy btw, if you want to see the same derivation of the product rule done a bit more clearly, see http://ocw.mit.edu/resources/res-18-006-calculus-revisited-single-variable-calculus-fall-2010/part-ii-differentiation/lecture-1-derivatives-of-some-simple-functions/ it starts at about 13:00 minutes in.