I am looking for a function of exponential order(i.e. bounded in absolute value by $Me^{ct}$ for some $M, c$) whose derivative is not of exponential order.
My thought was to look for functions which oscillate wildly between + and - in continually decreasing "periods," but I'm having trouble finding specific functions. Any help is appreciated.
How about this: $\sin(e^{t^2})$