In article <firstname.lastname@example.org>, Anonymous <email@example.com> wrote:
> Keeping in mind df(x)/dx = f'(x), if y'(x) = 1/x'(y) and, > likewise, x'(y) = 1/y'(x), how do you express 1/y'(x) > in terms of "d"? > Is it just dx/dy(x) ....
Leibniz was very concerned about good notation. He tried and discarded various notations before settling on the d and long s. One of the beautiful mnemonics built into that notation is dx/dy = 1/(dy/dx). It looks just like manipulating an algebraic fraction, although of course it's actually deeper. The same goes for the chain rule dz/dx = (dz/dy)(dy/dx), and the rule for integration by substitution ("cancel the dx").
It's always sad to see elementary calculus text-books which rigorously insist on f(x) and f'(x) all the time, denying students the gift which Leibniz gave us.