Question

in the proof for the quotient rule, it gets to the point where the lecturer divides "delta" u/v by "delta" x. how is it that he only divides the elements in the numerator.

Answer 1

because a*(1/b)*(1/c) = a*(1/c)*(1/b).

replace:
a with the numerator
b with the denominator
c with delta x

and you get what he got