Differentiation of a constant

A function that is known to have the same value for every value of the independent variable is constant, and we may denote it by

$$y = c.$$

As $x$ takes on an increment $\Delta x$, the function does not change in value, that is, $\Delta y = 0$, and

$$\frac{\Delta y}{\Delta x} = 0.$$

But

$$\lim_{\Delta x \to 0} \left ( \frac{\Delta y}{\Delta x} \right ) = \frac{dy}{dx} = 0.$$

Therefore, $\frac{dc}{dx} = 0$ (equation (I) above). The derivative of a constant is zero.