Derivative of 1/1+x^2

Learn what is the derivative of 1/1+x^2 with formula. Also understand how to verify the derivative of a function 1/1+x^2 by using first principle and power rule.

Alan Walker-

Published on 2023-06-27

Introduction to the derivative of 1/1+x^2

Derivatives have a wide range of applications in almost every field of engineering and science. The 1/1+x^2 derivative can be calculated by following the rules of differentiation. Or, we can directly find the derivative formula of 1/1+x^2 by applying the first principle of differentiation. In this article, you will learn what the derivative of 1/x^2 + 1 formula is and how to calculate the derivatives of 1/1+x^2 by using different approaches.

What is the derivative of 1/1+x^2?

The derivative of 1/1+x^2 with respect to the variable x is equal to 2x/(1+x^2)^2. It measures the rate of change of the algebraic function 1/1+x^2. It is denoted by d/dx(1/1+x^2) which is a fundamental concept in calculus. Knowing the formula for derivatives and understanding how to use it can be used in solving problems related to velocity, acceleration, and optimization. 

1/1+x^2 derivative formula

The formula for derivative of f(x)=1/1+x^2 is equal to the 2x/(1+x^2)^2, that is;

$f'(x) = \frac{d}{dx}(\frac{1}{1+x^2}) = \frac{2x}{(1+x^2)^2}$

It is calculated by using the power rule of derivatives, which is defined as:

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

How do you differentiate 1 by 1+x2?

There are multiple ways to prove the differentiation of 1/1+x^2. These are;

  1. First Principle
  2. Power rule
  3. Quotient Rule

Each method provides a different way to compute the 1/1+x^2 differentiation. By using these methods, we can mathematically prove the formula for finding the differential of 1/1+x^2.

Derivative of 1/1+x^2 by first principle

According to the first principle of derivative, the 1/1+x^2 derivative is equal to -2x/(1+x^2)^2. The derivative of a function by first principle refers to finding a general expression for the slope of a curve by using algebra. It is also known as the delta method. The derivative is a measure of the instantaneous rate of change, which is equal to,

$f’(x) = \lim_{h\to 0} \frac{f(x + h) - f(x)}{h}$

This formula allows us to determine the rate of change of a function at a specific point by using the limit definition of derivative calculator.

Proof of 1 by 1+x^2 derivative formula by first principle

To prove the 1/1+x^2 derivative by using first principle, replace f(x) by 1/1+x^2 or you can replace it by 1/1+x^2 to find the derivative. 

$f’(x) = \lim_{h→0}\frac{f(x + h) - f(x)}{h}$

$f’(x) = \lim_{h \to 0} \frac{1/1+(x + h)^2 - 1/1+x^2}{h}$


$f’(x) = \lim_{h \to 0} \frac{(1+x^2)-(1+x^2+h^2+2xh)}{h(1+x^2)(1+(x+h)^2)}{2}nbsp;

$f’(x) = \lim_{h \to 0} \frac{-h^2-2xh}{h(1+x^2)(1+(x+h)^2)}$


$f’(x) = \lim_{h \to 0} \frac{-h-2x}{(1+x^2)(1+(x+h)^2)}$

When h approaches to zero,

$f’(x) = \frac{-2x}{(1+x^2)^2}$

Hence the differentiation of 1/1+x^2 is equal to -2x/(1+x^2)^2. Also calculate the derivative of x^2 graph by using our derivative graph calculator.

Derivative of 1/1+x^2 by power rule

The formula for derivative 1 by 1+x^2 can be calculated by using the power rule because it is used to calculate derivatives of algebraic functions with some power. The power rule derivative is defined as;


Proof of differentiating of 1/1+x^2 by power rule

To differentiate of 1/1+x^2 by using product rule, we start by assuming that,

$f(x) =\frac{1}{1+x^2}=(1+x^2)^{-1}{2}nbsp;

By using power rule of differentiation,

$f’(x) = (-1)(1+x^2)^{-1-1}\frac{d}{dx}(x^2)$

We get,

$f’(x) =(-1)(1+x^2)^{-2}(2x){2}nbsp;

where the derivative of x squared is 2x.


$f’(x) = \frac{-2x}{(1+x^2)^2}$

Also, this method can be used to solve the differentiation of x^4.

Derivative of 1/x^2+1 using quotient rule

Another method for finding the derivative of 1/x^2 + 1 is using the quotient rule, which is a formula for finding the derivative of a quotient of two functions. Since the secant function is the reciprocal of cosine, the derivative of cosecant can also be calculated using the quotient rule. The derivative quotient rule is defined as:

$\frac{d}{dx}(f/g) = \frac{f’(x).g(x) - g’(x).f(x)}{(g(x))^2}$

Proof of differentiating x by quotient rule

To prove the x^2 derivative, we can start by writing it,

$f(x) =\frac{1}{1+x^2} = \frac{u}{v}$

Supposing that u = 1 and v = 1+x^2. Now by quotient rule,

$f’(x) =\frac{v.u’ - uv’}{v^2}$

$f’(x) = \frac{(1+x^2)(1)’ - (1)(1+x^2)’}{(1+x^2)^2}$

$f’(x)=\frac{(1+x^2)(0) - (1)(2x)}{(1+x^2)^2}{2}nbsp;


Hence, we have derived the derivative of 1/x2+1 using the quotient rule of differentiation. Use our quotient rule calculator to find the derivative of any quotient function.

How to find the derivatives of 1/1+x^2 with a calculator?

The easiest way of differentiating 1/1+x^2 is by using a dy/dx calculator. You can use our derivative calculator for this. Here, we provide you a step-by-step way to calculate derivatives by using this tool.

  1. Write the function as 1/1+x^2 in the enter function box. In this step, you need to provide input value as a function that you want to differentiate.
  2. Now, select the variable by which you want to differentiate 1/1+x^2. Here you have to choose x.
  3. Select how many times you want to calculate the derivatives of 1/1+x^2. In this step, you can choose 2 for second, 3 for triple differentiation and so on.
  4. Click on the calculate button. After this step, you will get the derivative of 1/1+x2 within a few seconds.
  5. After completing these steps, you will receive the differential of x square within seconds. Using online tools can make it much easier and faster to calculate derivatives, especially for complex functions.


In conclusion, the derivative of 1/1+x2 is -2x/(1+x^2)^2. The derivative measures the rate of change of a function with respect to its independent variable, and in the case of 1/1+x^2, the derivative can be calculated using the power rule, product rule and quotient rule of differentiation. By applying this rule, we find that the derivative of 1/1+x^2 is -2x/(1+x^2)^2, indicating that the rate of change of 1/1+x^2 increases linearly with the value of x. 

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