Constant rule in calculus

Statement

The derivative of a constant is always $0$.

$$\tfrac{d}{dx}(c) = 0$$

Proof

Define function $f$.

$$f(x) = c$$

$$f'(x) = \lim_{h \to 0}\left(\frac{f(x + h) - f(x)}{h}\right)$$

Substitue $f(x) = c$ and $f(x + h) = c$.

$$f'(x) = \lim_{h \to 0}\left(\frac{c - c}{h}\right)$$

Finally, simplify.

$$f'(x) = \lim_{h \to 0}\left(\frac{0}{h}\right)$$

$$f'(x) = \lim_{h \to 0}\big(0\big)$$

$$f'(x) = 0$$