Leibniz notation, need clarification

[Werg22]

This question comes from how Leibniz chose his notation.

How to prove that the limit when h goes to 0 of the expression:

$$\frac{f(x + 2h) - f(x + h) - [f(x + h) - f(x)]}{h^{2}}$$

is f''(x)?


Step 1: We know that

$$\frac{f(x + h) - f(x)}{h} = f'(x) + a$$

Where "a" is a value that can be as small as we want, in function of h.

Step2:

Also,

It has occurred to me that first we must prove that

$$\frac{f(x + 2h) - f(x + h)}{h}$$ can be written under the form

$$f'(x + h) + b$$

Step 3: The last condition that must be fufilled is that the limit as h goes to 0 of the expression $$\frac{b - a}{h}$$ is 0.

Step 4: That way we start with

$$\frac{f(x + 2h) - f(x + h) - [f(x + h) - f(x)]}{h^{2}}$$

We write

$$\frac{f'(x + h) + b - [f'(x) + a]}{h}$$

We rearange so

$$\frac{f'(x + h) - f'(x)}{h}+ \frac{b - a}{h}$$

Now it would be clear the limit is f''(x).

The real problem is to prove step 2 and step 3... I tried but nothing occurred to me. Anyone care to try/help? Thanks in advance.

 

[Data]

Assuming that $f$ is twice differentiable (which you must be already), just applying l'Hopital's rule twice to the original expression will get the result.

 

[Werg22]

Okay, I see. Did Leibniz had the notions of what Hopital's rule implies? If not, there must be another evidence that Leibniz fell upon. Any idea in that case?

 

[arildno]

You can't fantasize in this way about the historical evolution of notation, Werg22!

To give you just a hint:
The notation f(x) was not at all developed at the time Leibniz chose his notation!
In fact, it was Euler, about 100 years after Leibniz who developed a proto-notation that eventually developed into f(x) in the 19th century.

Furthermore, proofs were generally something very different in the 17th century from proofs of today.

Apart from some solid arguments, most accepted proofs at that time is regarded as mere hand-waving in our time.


There are too many ways that a notation COULD have developed, making it IMPOSSIBLE to deduce how it actually came about.
The only way to find this out is to read the actual works by Leibniz, or better, works by competent commentators on Leibniz.