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