Derivative of x is 1

The derivative of x is 1 (one). Now by the definition or first principle we shall show that the 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 the limit of both sides as $$\Delta x \to 0$$, we have
\[ \Rightarrow \mathop {\lim }\limits_{\Delta x \to 0} \frac{{\Delta y}}{{\Delta x}} = \mathop {\lim }\limits_{\Delta x \to 0} \left( 1 \right)\]

The limit does not affect 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
\[y = 7x\]

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