# Derivative of 1/x: Proof by First Principle

The derivative of 1/x is equal to -1/x2. In this blog post, we will learn how to find the differentiation of 1/x.

Derivative of 1/x Formula: The formula of the derivative of 1/x is given by

d/dx(1/x) = -1/x2.

Let us find the differentiation of 1/x by first principle which is equal to -1/x2.

## Derivative of 1/x using First Principle

By the first principle, the derivative of a function f(x) is given by the following limit formula:

$\dfrac{d}{dx}(f(x))$ = limh→0 $\dfrac{f(x+h)-f(x)}{h}$

Let f(x) = 1/x in the above limit. So the derivative of 1/x from the first principle will be equal to

∴ $\dfrac{d}{dx}(1/x)$ $=\lim\limits_{h \to 0} \dfrac{\dfrac{1}{x+h}-\dfrac{1}{x}}{h}$

= limh→0 $\dfrac{x-(x+h)}{hx(x+h)}$

= limh→0 $\dfrac{x-x-h}{hx(x+h)}$

= limh→0 $\dfrac{-h}{x(x+h)}$

= $\dfrac{-1}{x(x+0)}$

= $-\dfrac{1}{x\cdot x}$

= -1/x2

Thus, the derivative of 1/x is equal to -1/x2 and this is obtained by the first principle of derivatives.

## What is the Derivative of 1/x?

The derivative of 1/x can be obtained by using the power rule of derivatives. Recall, the power rule of derivatives:

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

First, write $1/x$ as follows:

$\dfrac{1}{x} = x^{-1}$

Differentiate both sides with respect to x. So we obtain that

$\dfrac{d}{dx}(\dfrac{1}{x}) = \dfrac{d}{dx}(x^{-1})$ = -1 × x-1-1 = -1/x2.

So the derivative of 1/x by the power of derivatives is equal to -1/x2.

## FAQs

Q1: What is the derivative of 1/x?

Answer: The derivative of 1/x is equal to -1/x2.

Q2: Find the differentiation of 1/x.

Answer: The differentiation of 1/x is -1/x2.