# 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:

Putting the value of function $y = c$ in the above equation, we get

Dividing both sides by $\Delta x$, we get

Taking the limit of both sides as $\Delta x \to 0$, we have

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

Differentiating with respect to variable $x$, we get

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