2

We know that $Ce^x$ and $0$ are the two functions whose first derivative is equal to itself, but what about derivatives of a higher order? For example, the second derivative of $e^{-x}$ is equal to itself, but not the first, and the fourth derivative of $sin(x)$ is equal to itself.

In short, are there other examples of functions whose nth derivative is equal to itself, where $n>1$?

Thank you kindly!

Gizmo
  • 725
  • 2
    see this discussion http://math.stackexchange.com/questions/1646912/which-derivatives-are-eventually-periodic – reuns Jun 15 '16 at 00:35

1 Answers1

2

The second derivative of $ce^x+ke^{-x}$ is the same as the original function, same goes for $\cosh(x)$ and $\sinh(x)$.

The third derivative of $e^{\omega x}$ is the same as the original function.

Colbi
  • 823