📂Calculus

Differentiation of Constant Functions

Formula

The derivative of the constant function $C$ is $0$.

$$\dfrac{d C}{dx} = 0$$

Explanation

To be precise, the derivative being a function means “the derivative of the constant function is the zero function.” Since the zero function is also a constant function, the derivative of the constant function is a constant function.

Derivation

For all $x \in \mathbb{R}$, let it be $C(x) = c$ ($c \in \mathbb{R}$ is an arbitrary constant). According to the definition of derivative,

$$\begin{align*} \dfrac{d C(x)}{dx} &= \lim_{h \to 0} \dfrac{C(x+h) - C(x)}{h} \\ &= \lim_{h \to 0} \dfrac{c - c}{h} \\ &= \lim_{h \to 0} 0 \\ &= 0 \end{align*}$$

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