Notation for the derivative function as a limit using the second definition of the derivative of a function at a point

Question

Given a function $f(x)$, its derivative at the point $c$ is defined as the following limit:

$$f'(c) = \lim_{h\to 0}\frac{f(c+h)-f(c)}{h} = \lim_{\Delta x\to 0}\frac{f(c+\Delta x)-f(c)}{\Delta x} \tag{1}$$

And if we make the change of variables $c+h=x$, then:

$$f'(c) = \lim_{x\to c}\frac{f(x)-f(c)}{x-c} \tag{2}$$

On the other hand, the derivative function $f'(x)$ is defined, replacing at $(1)$ $c$ by $x$:

$$f'(x) = \lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=\lim_{\Delta x\to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x} =\lim_{\Delta x\to 0}\frac{\Delta f(x)}{\Delta x}\tag{3}$$

Then, which would be the notation for the derivative function making use of the second definition of the derivative $(2)$?

Everything you've written looks fine to me, although the very last bit of (3) is uncommon notation I think. I don't understand the question. — Adam Rubinson, Dec 23 '19 at 18:33
Ah, is the question: "what is the definition of the derivative function?" — Adam Rubinson, Dec 23 '19 at 18:34
The notations available for any of the definitions are the same, and there are quite a few. — Adrian Keister, Dec 23 '19 at 20:58

score 1 · Answer 1 · answered Dec 23 '19 at 18:29

1

You could write

$$f'(x) = \lim_{\xi\to x}\frac{f(x)-f(\xi)}{x-\xi}$$

answered Dec 23 '19 at 18:29

Quaerendo

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Notation for the derivative function as a limit using the second definition of the derivative of a function at a point

1 Answers1