Derivative of x by 1+x

Learn what is the derivative of x/1+x with formula. Also understand how to verify the derivative of an algebraic function x/1+x by using first principle.

Alan Walker-

Published on 2023-06-28

Introduction to the derivative of x/1+x

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

What is the derivative of x/1+x?

The derivative of x/1+x with respect to the variable x is equal to 1/(1+x)^2. It measures the rate of change of the algebraic function x/1+x. It is denoted by d/dx(x/1+x) 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. 

Derivative of x/1+x formula

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

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

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+x?

There are multiple derivative laws to prove the differentiation of x/1+x. These are;

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

Each method provides a different way to compute the x/1+x differentiation. By using these methods, we can mathematically prove the formula for finding the differential of x/1+x.323919

Derivative of x/1+x by first principle

According to the first principle of derivative, the x/1+x derivative is equal to 1/(1+x)^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 x by 1+x derivative formula by first principle

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

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

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


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

Derivative of x/1+x by product rule

Another method to find the derivative x by 1+x is the product rule formula which is used in calculus to calculate the derivative of the product of two functions. Specifically, the product rule is used when you need to differentiate two functions that are multiplied together. The formula for the product rule is:

d/dx(uv) = u(dv/dx) + (du/dx)v 

In this formula, u and v are functions of x, and du/dx and dv/dx are their respective derivatives with respect to x.

Proof of differentiating of x/1+x by product rule

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

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

By using product rule of differentiation,

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

We get,

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

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

After simplification, we get

$f’(x) = \frac{-x+1+x}{(1+x)^2}$


$f’(x) = \frac{-1}{(1+x)^2}$

Also use our derivative product rule calculator online, as it provides you a step-by-step solution of differentiation of a function.

Derivative of 1/x+1 using quotient rule

Another method for finding the differential of x/1+x 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 quotient rule of derivative 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,

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

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

$f’(x) =\frac{v.u’ - uv’}{v^2}$

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

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


Hence, we have derived the derivative of x/x+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 x/1+x with a calculator?

The easiest way of differentiating x/1+x 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 x/1+x 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 x/1+x. Here you have to choose x.
  3. Select how many times you want to calculate the derivatives of x/1+x. In this step, you can choose 2 to calculate the second derivative, 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 by 1+x within seconds. Using online tools can make it much easier and faster to calculate derivatives, especially for complex functions.


In conclusion, the derivative of x/1+x is 1/(1+x)^2. The derivative measures the rate of change of a function with respect to its independent variable, and in the case of x/1+x, 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 x/1+x is 1/(1+x)^2, indicating that the rate of change of x/1+x increases linearly with the value of x. 

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