# Is there a “natural tetration function”?

For the natural exponential function,

$$f(x)=e^x \to f(x)=f'(x).$$

Is there a natural tetration (tetral?) function?

$$f(x)={{^x}b} \to f(x)=f'(x)$$

Is it base $$e$$?

• You seem to be confusing two things - the differential equation $f'(x)=f(x)$ has the solution $f(x)=ae^x$ for a constant $a$, so you won't get anything essentially new from that. (consider $e^{-x}f(x)$ and differentiate this to obtain $-e^{-x}f(x)+e^{-x}f'(x)=0$ so $e^{-x}f(x)=a$ and $f(x)=ae^x$) – Mark Bennet Jun 1 at 7:54
• I see it now. Over morning coffee I did feel it would be nice if a natural tetration function did exist. But thanks for clarifying. – lukejanicke Jun 1 at 8:20
• Is there anything natural about tetration? – darij grinberg Jun 1 at 9:18

$$f(x)=f'(x)$$ implies that $$f(x)=ce^x$$, so an exponential function, not tetration.