The differentiation of 1/x with respect to x is equal to -1/x^{2}. In this post, we will learn how to differentiate 1 by x.

## Derivative of 1/x by Power Rule

**Question:** What is the Derivative of 1/x?

**Answer:** The derivative of 1/x with respect to x is -1/x^{2}.

**Solution:**

By the power rule, the derivative of of x raised to the power n is given by

d/dx (x^{n}) = nx^{n-1}.

As 1/x can be written as x^{-1}, so by the above rule, its derivative is equal to

d/dx (1/x) = d/dx (x^{-1}) = -1 (x^{-1-1}) = -1/x^{2}.

So the derivative of 1/x by power rule is equal to -1/x^{2}.

**Read Also:** Derivative of e^{sinx}

## Applications

**Question 1:** Find the derivative of 1/x+1.

**Answer:**

Let z=x+1. Then dz/dx =1.

By the chain rule of derivatives,

$\dfrac{d}{dx}(\dfrac{1}{x+1})$ = $\dfrac{d}{dz}(\dfrac{1}{z}) \times \dfrac{dz}{dx}$

= $-\dfrac{1}{z^2} \times 1$

= $-\dfrac{1}{(x+1)^2}$ as z=1+x.

So the derivative of 1/1+x is equal to -1/(1+x)^{2}.

## FAQs

### Q1: What is the differentiation of 1/x?

Answer: The differentiation of 1/x is equal to -1/x^{2}.