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?