Let's start the simplest place there is. A constant function always returns the same number, no matter what you put in:

Its graph is a perfectly flat horizontal line at height c. Nothing about the output changes as x moves — so the question we always ask of a derivative, how fast is the output changing?, has an easy answer here: it isn't.

A flat line has slope zero

The derivative measures slope — the steepness of the curve at a point. Walk along a flat road and you climb nothing: rise 0 over any run. So the slope of f(x) = c is 0 everywhere. Drag the point along the line below and read its slope off live — it never budges from zero.

The same answer from the difference quotient

We can prove it instead of just seeing it. The difference quotient measures the average rate of change over a tiny step h:

Both outputs are the same constant c, so the top is c - c = 0. The ratio is 0 before we even shrink h, so the limit is 0 too. That gives our very first rule:

See it together

On the left is the constant function f(x) = 2; on the right its derivative f'(x) = 0. The function sits at a fixed height, and its slope is flat on the axis — zero at every input.