TITLE: Definition of the Derivative Using a Sequence
QUESTION [6 upvotes]: I believe that I may have once seen a definition of the derivative that went along these lines:

$$f'(c)=\lim_{n\to\infty}\frac{f(x_n)-f(c)}{x_n-c}$$

Here, $(x_n)$ is a sequence that converges to $c$.
As I do not recall much about it, I may have left out one or two important assumptions. However, does this seem in any way correct? And if so, does this theorem/etc. have a name? Thanks in advance!

REPLY [7 votes]: One might say that the derivative $f'(x)$ exists at $c$ if and only if $\lim \dfrac{f(x_n) - f(c)}{x_n - c}$ exists and is the same for any sequence $x_n$ s.t. $x_n \to c$
If you parse what this means, you can see that this is equivalent to the standard definition. In particular, it's easy to see that if a function isn't differentiable under this definition, then it's not in the standard, and vice versa.