Numerical Differentiation: Comparing Central & Other Formulas

[t0mm02]

Homework Statement

Central, Forward, and Backward difference equations.

Relevant Equations

Central, Forward, and Backward difference equations.

Hello there, I have found a different central differentiation formula for a first derivate from what I am used to seeing and I was wondering if they were the same one. I am struggling to find the Numerical Differentiation formulas (forward, backward and central) in scholarly articles and I have found one with the supposed formulas but they look so different from how I normally see them.
This is the formula I have seen:

This is the formula I normally see:

 

[Stephen Tashi]

This is the formula I have seen:
View attachment 276503

I'm not sure what the $d$ symbol denotes, but this looks like a formula for a "multipoint" approximation of $f'$. For example, the slide show https://www3.nd.edu/~gtryggva/CFD-Course/2011-Lecture-8.pdf gives examples of multipoint approximations in simpler notation.