How is it that you're allowed to treat the derivative dy/dx as an actual quotient, f.ex. when calculating in integral by substitution such as: integral f(x) dx, if you the y=f(x), and dy/dx = g(x), so that dx = dy/g(x) and so integral f(x) dx = integral y/g(x) dy.
Clearly, we multiplied both sides of dy/dx = g(x) with dx! How is this possible? Isn't dy/dx just a notation denoting the derivative, not actual.... numbers?