 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?
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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.