Usually f'(x) denotes the differential of f(x), doesn't it?

Prime doesn't necessarily need to denote anything specific, but often means 'different, but related'. It is common to use prime notation to reflect a derivative function in calculus, but like most things it depends on context.

It was Lagrange's notation, I believe. As opposed to Leibniz's notation which is the dy/dx notation that is all the rage now.

Now that's a cool accident... thanks for sharing :)