Question

Why is the derivative of a constant zero ​

Answer 1

A constant function is exactly that - constant - meaning it exhibits no change whatsoever with respect to any change in its input. If $f(x) = 1$, then it doesn't matter what value of $x$ I give you, the value of $f(x)$ will always be nothing other than 1.

That the derivative of such a function is zero follows immediately from the definition of the derivative. If $c\in\Bbb R$ and $f(x) = c$, then

$f'(x) = \displaystyle \lim_{h\to0}\frac{f(x+h)-f(x)}h = \lim_{h\to0} \frac{1 - 1}h = \lim_{h\to0}\frac0h = 0$