Derivative of a binary number:

Think of any binary number as a sequence of 0s and 1s in a certain order

For example, 16 in binary is the sequence: 1000

Reading the sequence from right to left, two bits at a time, for each one of those two bits, we can note if the value of the “earlier” bit in the sequence changed

I can note a change as 1 and a not change as 0, then the above sequence becomes:

0 (0-0 no change) 0 (0-0 no change) 1 (0-1 changed)

Result: 100

In decimal: 8

Now if I want to “integrate” that sequence, I can do the reverse, but now I have ambiguity, if I start with 0, the sequence would be the original:

0 0 (0 means no change) 0 (0 means no change) 1 (1 change)

Result: 1000

But if we start with 1 instead:

1 1 (0 no change) 1 (0 no change) 0 (1 change)

Result: 0111

Intuitively you can think of this as tracking a “discrete rate of change”

Usually the derivative or slope of a function gives a real value, now imagine “zooming into” the function until you can’t track a real value anymore, only whether what you are looking at is changing or not every time you look