I am a STUDENT in 11th grade who has just finished BC Calculus.

I don't have a PhD or even a high school diploma, obviously.

But I think that beginning with h->0 is essential. Otherwise, we don't have any definition of a derivative.

But more to the point, I don't think it's strictly necessary to learn the concepts completely before you do symbolic calculations.

How would the teacher make sure that the students eventually learn them before it's too late, then: Leibniz's lovely notation.

Newton's notation is essentially a meaningless shorthand. Prime is arbitrarily chosen to mean a derivative. I don't like that. (I like it as a shorthand, but there is no real meaning behind it).

But Leibniz's has actual meaning: dy/dx is analogous to delta-y/delta-x.

If we are always used to writing dy/dx=... or df(x)/dx=..., then it is no great stretch to write things like df(x)=...dx. And this leads us nicely into differential equations by separation of variables, and concepts such as substituting variables when you integrate.

In my humble opinion, using Newton's notation should be avoided as much as possible, because it doesn't make it clear what you are doing and turns people into robots, mindlessly following the rules of differentiation.

I don't think that Newton's method id all bad. I think it may be good when you are taking derivatives of higher orders, because once you have the concept of derivatives down, it's more important to see that you are taking the derivative of another derivative. (The "exponents" in Leibniz's notation make it a bit confusing).

If I were a Calculus teacher (and I very well may become one someday), I would all but scrap Newton's notation.