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
\[\begin{gathered} \frac{{dy}}{{dx}} = 0 \\ \Rightarrow \frac{d}{{dx}}\left( 9 \right) = 0 \\ \end{gathered} \]