Can you differentiate a number and get a nonzero answer? You can't with the usual kind of differentiation, but you can with the “arithmetic derivative”. The basic version of the arithmetic derivative takes positive integers as inputs. The only requirements are the two shown in the picture: (1) the derivative of a prime number is 1, and (2) the product rule holds. It can be proved fairly easily that there is only one function satisfying these requirements. The table shows the values of the arithmetic derivatives of the first few positive integers; these values will always be other nonnegative integers.
The arithmetic derivative
The arithmetic derivative
The arithmetic derivative
Can you differentiate a number and get a nonzero answer? You can't with the usual kind of differentiation, but you can with the “arithmetic derivative”. The basic version of the arithmetic derivative takes positive integers as inputs. The only requirements are the two shown in the picture: (1) the derivative of a prime number is 1, and (2) the product rule holds. It can be proved fairly easily that there is only one function satisfying these requirements. The table shows the values of the arithmetic derivatives of the first few positive integers; these values will always be other nonnegative integers.