Derivative of a constant with respect to x


$\dfrac{d}{dx} \, c = 0$


The differentiation of a constant term can be derived in mathematics on the basis of the mathematical relation of the differentiation with limits.

$$\dfrac{d}{dx} f(x) = \lim_{h \to 0} \dfrac{f(x+h)-f(x)}{h}$$

Assume $f(x) = c$. There is no $x$ term in this function and it is constant for everything. Hence, $f(x+h) = c$. Substitute both functions in the fundamental law of the differentiation.

$$\implies \dfrac{d}{dx} \, c = \lim_{h \to 0} \dfrac{c-c}{h}$$

There are two constants in numerator but they both are same algebraically. Hence, the subtraction of them is zero.

$$\implies \dfrac{d}{dx} \, c = \lim_{h \to 0} \dfrac{0}{h}$$

$$\implies \dfrac{d}{dx} \, c = \lim_{h \to 0} 0$$

There is no $h$ term. So, the limit can be removed from operation.

$\therefore \,\,\,\,\,\, \dfrac{d}{dx} \, c = 0$

The differentiation formula has proved that the derivative of any constant term with respect to $x$ is zero.

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