here is one proof that I know but I am not totally sure if it is acceptable-
exponential functions are exponential: no matter how many times you differentiate them e.g-
first derivative f`(x)= e^x
2nd derivative f``(x)= e^x
3rd derivative f```(x)=e^x
and so on.
now if you differentiate a polynomial function- let's say,
1st derivative f`(x)= 5x^4
2nd de3rivative f``(x)= 20x^3
3rd derivatives f```(x)=60x^2
4th derivative f````(x)=120x
5th derivative f`````(x)= 0
like this every polynomial finally gets differentiated to zero or a constant . this proves that the polynomials are not exponential.
**is my proof ok**
I want more alternate proofs and a brief explanation about this one.