Differentiation of 1/x | Differentiate 1/x

The differentiation of 1/x with respect to x is equal to -1/x². 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/x².

Solution:

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

d/dx (xⁿ) = 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/x².

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

Read Also: Derivative of e^sinx

Differentiation of e^cosx

Differentiation of e^tanx

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.

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