# The Derivative of a Constant Function

The derivative of a constant function is zero. Now we shall prove this constant function with the help of the definition of derivative or differentiation.

Let us suppose that $y = f\left( x \right) = c$ where $c$ is any real constant.

First we take the increment or small change in the function:
$\begin{gathered} y + \Delta y = c \\ \Rightarrow \Delta y = c – y \\ \end{gathered}$

Putting the value of function $y = c$ in the above equation, we get
$\begin{gathered} \Rightarrow \Delta y = c – c \\ \Rightarrow \Delta y = 0 \\ \end{gathered}$

Dividing both sides by $\Delta x$, we get
$\begin{gathered} \frac{{\Delta y}}{{\Delta x}} = \frac{0}{{\Delta x}} \\ \Rightarrow \frac{{\Delta y}}{{\Delta x}} = 0 \\ \end{gathered}$

Taking the limit of both sides as $\Delta x \to 0$, we have
$\begin{gathered} \Rightarrow \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\Delta y}}{{\Delta x}} = \mathop {\lim }\limits_{\Delta x \to 0} \left( 0 \right) \\ \Rightarrow \frac{{dy}}{{dx}} = 0 \\ \Rightarrow \frac{d}{{dx}}\left( c \right) = 0 \\ \end{gathered}$

This shows that the derivative of a function is zero.

Example: Find the derivative of $y = f\left( x \right) = 9$

We have the given function as
$y = 9$

Differentiating with respect to variable $x$, we get
$\frac{{dy}}{{dx}} = \frac{d}{{dx}}\left( 9 \right)$

Now using the formula for a constant function $\frac{d}{{dx}}\left( c \right) = 0$, we have
$\begin{gathered} \frac{{dy}}{{dx}} = 0 \\ \Rightarrow \frac{d}{{dx}}\left( 9 \right) = 0 \\ \end{gathered}$