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`