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.

Also Read: 

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)

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