# Derivative of x by First Principle, Power Rule

The derivative of x is equal to 1. The function f(x)=x denotes the identity function on the set of real numbers. In this post, we will find the derivative of x by different methods; for example, by power rule, by the limit definition of derivatives.

## Derivative of x by Power Rule

We will use the power rule of derivatives to find the derivative of x. The power rule tells us that the derivative of x to the power n is as follows:

$\dfrac{d}{dx}(x^n)=nx^{n-1}$

Put n=1 in the above rule. Then the derivative of x is equal to

$\dfrac{d}{dx}(x)=1 \cdot x^{1-1}$ $=x^0$ $=1$ as we know that any element to the power zero is 1.

Hence, the derivative of x is 1 and this is obtained by the power rule of derivatives.

Now, from the first principle, that is, using the limit definition of derivatives, we will evaluate the derivative of x.

## Derivative of x by First Principle

Let f(x)=x. Then the derivative of f(x) by the first principle is given as follows.

$\dfrac{d}{dx}(f(x))$ $=\lim\limits_{h \to 0} \dfrac{f(x+h)-f(x)}{h}$

As f(x)=x, so the derivative of x from the first principle will be

$\dfrac{d}{dx}(x)$ $=\lim\limits_{h \to 0} \dfrac{(x+h)-x}{h}$

$=\lim\limits_{h \to 0} \dfrac{x+h-x}{h}$

$=\lim\limits_{h \to 0} \dfrac{h}{h}$

$=\lim\limits_{h \to 0} 1$

$=1$.

Hence, the derivative of x by first principle is 1.

Notes on Derivative of x:

• The function f(x)=x is the identity function, and its derivative is 1.
• As x=x1, by power rule, the derivative of x is 1.

Derivative of 1/root(x)

Derivative of x2

Derivative of root x + 1 by root x

Derivative of 1/x2

Derivative of 1/(1+x2)

## FAQs

Q1: What is the derivative of x?

Answer: The derivative of x is equal to 1 and it can be proved by the first principle of derivatives as well as the power rule of derivatives.