# The Prime Derivative

Given a positive integer $$x$$, define the derivative of $$x$$, denoted, $$x’$$ as:

$$x’ = 1$$ if $$x$$ is a prime.

$$x’={a}\cdot{b’} + {a’}\cdot{b}$$ if $$x={a}\cdot{b}$$ ($$x$$ is composite)

$$x’=0$$ if $$x=1$$

Since all composites are products of primes, we can find the derivatives of all orders for all integers.  What patterns are observed?