Why is the derivative of constant zero?

The derivative of a constant is zero because a constant does not change, and the derivative measures rate of change.

Definition-Based Explanation:

The derivative of a function f(x) at a point is defined as:

f'(x) = lim(h → 0) [f(x+h) - f(x)] / h

If f(x) = c (a constant):

• f(x + h) = c
• f(x) = c

Then:

f'(x) = lim(h → 0) [c - c] / h = lim(h → 0) 0/h = 0

Visual Explanation:

A constant function like f(x) = 5 is a horizontal line. The slope of any horizontal line is 0. Since the derivative gives the slope of the tangent at any point, the derivative of a constant is zero.