Second Derivatives to Second Differences

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In these two videos...

https://www.youtube.com/watch?v=BLkz5LGWihw

https://www.youtube.com/watch?v=ly4S0oi3Yz8

...Grant shows how second derivative of $f(x)$ at some point $(x_0)$ are related to the "second differences" of the functoin.

Mathematically written:

$$[f(x_0 +dx) - f(x_0)]+[f(x_0-dx)-(f(x_0))] \propto \frac{d^2f}{dx^2}(x_0)$$

I'm having a little bit of trouble understanding this...if someone could please explain it, I'd really, really appreciate it.

Thank you!