I think you mean to put $$dL^2=dx^2+dy^2+dz^2\\dL^2=dr^2+r^2dφ^2+dz^2$$
For your proof, in a cylindrical coordinate system
and because of product rule
it follows that
and just substitute that for dx and find what dy is in terms of r and φ.
The expression $$d(f(x)g(x))=f(x)*dg(x)+g(x)*df(x)$$
is just the product rule. If you multiply both sides of the equation by 1/dx then it looks much more familiar.