Derivative of Constant Function

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

Differentiation with respect to variable x, we get

\frac{{dy}}{{dx}}  = \frac{d}{{dx}}\left( 9 \right)

Now using the formula for 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}