I am interested if there are any theorems/conjectures on the rates of change of transcendental functions. More specifically, I want to know that if $f(x)$ is a transcendental function, and $f(a)\in \Bbb Q$, if $f'(a)$ is transcendental. I know that there are exceptions such as $\exp'(0)=1$ and $\ln(x)$ but I am wondering if these are just exceptions.
2026-02-22 22:18:23.1771798703
Any theorems on the derivatives of transcendental functions?
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