Quote by Calculuser
In Physics, we know that what "Δx" means. (Δx=x[itex]_{2}[/itex]x[itex]_{1}[/itex]) It can also be Δx>0 or Δx<0.In Calculus, Leibniz Notation shows dx=Δx and dy≠Δy I got that part why it's so.But if "Δx" can be both Δx>0 and Δx<0, for dx and dy it must be the same (dx,dy>0 or dx,dy<0) right??
Finally, in my opinion.We use from right and from left derivative to recognize that the function is differentiable.If we have a differentiable function, don't need to do this step.Therefore, Leibniz thought that "dx" infinitesimal number (dx=[itex]\frac{1}{∞}[/itex]>0) and used it a differentiable funtion, so that he didn't need to approximate the funtion to observe whether it's differentiable or not and he admitted his notation to find derivative of a funtion which doesn't need to be observed whether it's differentiable or not.
Is that all right?

dx and dy mean deltas for sufficiently small dx. delta by itself does not imply the increasingly accurate estimation of dy from dx. The small d implies the existence of an unobserved  in some sense transcendental quantity  which is the scale factor that translates dx into dy. I imagine that this is Leibniz's idea of derivative. Perhaps Mathwonk an comment on that.