Derivative of a constant (Euler's number)

Since $e$ is Euler's number, it is a fixed real number (approximately $2.71828$) and does not depend on $x$ at all. The function $f(x) = e$ is therefore a constant function, no different in kind from $f(x) = 5$ or $f(x) = \pi$.

By the constant rule, $\frac{d}{dx}[c] = 0$ for any constant $c$. Applying this with $c = e$ gives $f'(x) = 0$.

Yes. Pi, like Euler's number, is a fixed real number that does not depend on x. Any constant function has a derivative of zero by the constant rule. So replacing e with pi changes nothing about the outcome; the derivative remains zero.