Iteration types of differentiation

The differentiation operation is a transformation of differentiable functions and it has different iteration types for different differentiable functions:

• Identity: $c_1e^x$
• Idempotent: $c_1e^x + c_2$
• Involution: $c_1e^x + c_2e^{-x}$

All functions of a certain iteration type come from solutions to first order linear ordinary differential equations such as $y'=y$, $y''=y$, and $y''=y'$. The partial ordering of iteration types helps us to understand these differential equations. In general, polynomials, exp, sin, cos, sinh, and cosh and other similar functions all produce a finite number of differentiations.