Derivative of x is 1

The derivative of x is 1 (one). Now by definition or first principle we shall show that derivative of x is equal to 1 .
Let us suppose that

y = f\left( x \right) = x

First we take the increment or small change in the function.

\begin{gathered} y + \Delta y = x + \Delta x \\ \Rightarrow \Delta y = x + \Delta x - y \\ \end{gathered}

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

\begin{gathered} \Rightarrow \Delta y = x + \Delta x - x \\ \Rightarrow \Delta y = \Delta x \\ \end{gathered}

Dividing both sides by \Delta  x, we get

\begin{gathered} \frac{{\Delta y}}{{\Delta x}} = \frac{{\Delta  x}}{{\Delta x}} \\ \Rightarrow \frac{{\Delta y}}{{\Delta x}} =  1 \\ \end{gathered}

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

  \Rightarrow \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\Delta y}}{{\Delta  x}} = \mathop {\lim }\limits_{\Delta x \to 0} \left( 1 \right)

Limit does not effect on constant values, so

\begin{gathered} \Rightarrow \frac{{dy}}{{dx}} = 1 \\ \Rightarrow \frac{d}{{dx}}\left( x \right) =  1 \\ \end{gathered}

This shows that the derivative of x is 1.

Example: Find the derivative of y = f\left( x \right) =  7x
We have the given function as

y =  7x

Differentiation with respect to variable x, we get

\begin{gathered} \frac{{dy}}{{dx}} = \frac{d}{{dx}}7x \\ \Rightarrow \frac{{dy}}{{dx}} =  7\frac{d}{{dx}}x \\ \end{gathered}

Now using the formula for constant function \frac{d}{{dx}}\left( x \right) = 1, we have

\begin{gathered} \frac{{dy}}{{dx}} = 7\left( 1 \right) \\ \Rightarrow \frac{d}{{dx}}\left( {7x}  \right) = 7 \\ \end{gathered}