Derivative of x/(x+1)
2026-02-28 06:11 Diff
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Last updated on September 9, 2025

We use the derivative of x/(x+1) as a tool to understand how this function changes with respect to x. Derivatives are fundamental in various applications like calculating profit or loss. This discussion will explore the derivative of x/(x+1) in detail.

What is the Derivative of x/(x+1)?

To find the derivative of x/(x+1), we use the quotient rule. The function x/(x+1) is differentiable within its domain, except where x+1=0. Key concepts include: 

Quotient Rule: Used for differentiating functions in the form of one function divided by another. 

Simplification: After applying the quotient rule, simplification of the resulting expression is necessary.

Derivative of x/(x+1) Formula

The derivative of x/(x+1) can be denoted as d/dx (x/(x+1)) or (x/(x+1))'.

The formula derived using the quotient rule is: d/dx (x/(x+1)) = 1/(x+1)²

This formula is valid for all x where x+1≠0.

Proofs of the Derivative of x/(x+1)

We derive the derivative of x/(x+1) using the quotient rule. To demonstrate this, we consider: -

Let u = x and v = x+1 

By the quotient rule: d/dx (u/v) = (v·du/dx - u·dv/dx) / v²

By applying the quotient rule: u = x, du/dx = 1 v = x+1, dv/dx = 1 d/dx (x/(x+1)) = [(x+1)·1 - x·1] / (x+1)² = (x+1 - x) / (x+1)² = 1/(x+1)²

Thus, the derivative of x/(x+1) is 1/(x+1)².

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Higher-Order Derivatives of x/(x+1)

Higher-order derivatives are obtained by differentiating the function multiple times. For x/(x+1), the first derivative is 1/(x+1)². The second derivative involves applying the chain rule or quotient rule again to 1/(x+1)². 

First derivative: f'(x) = 1/(x+1)² 

Second derivative: f''(x) = -2/(x+1)³

This pattern continues for higher derivatives.

Special Cases:

-At x = -1, the function is undefined, as the denominator becomes zero. 

At x = 0, the derivative is 1/(0+1)², which is 1.

Common Mistakes and How to Avoid Them in Derivatives of x/(x+1)

Common mistakes occur when differentiating x/(x+1). These can be avoided by understanding the correct methods. Here are some frequent errors and solutions:

Problem 1

Calculate the derivative of (x/(x+1))²

Okay, lets begin

Let y = (x/(x+1))².

Using the chain rule, dy/dx = 2(x/(x+1))·d/dx(x/(x+1))

We know d/dx(x/(x+1)) = 1/(x+1)².

Substitute into the equation: dy/dx = 2(x/(x+1))·(1/(x+1)²) = 2x/(x+1)³

Explanation

We find the derivative by first identifying the function's composition and applying the chain rule. Simplification leads to the final result.

Well explained 👍

Problem 2

A tank is filling with water, and its height is represented by h(x) = x/(x+1) meters after x minutes. Find the rate of change of height when x = 2 minutes.

Okay, lets begin

We have h(x) = x/(x+1).

Differentiate to find the rate of change: dh/dx = 1/(x+1)².

Substitute x = 2: dh/dx = 1/(2+1)² = 1/9

The rate of change of height at x = 2 minutes is 1/9 meters per minute.

Explanation

To find the rate of change at a specific time, we differentiate h(x) and evaluate the derivative at x = 2. This gives the instantaneous rate of change.

Well explained 👍

Problem 3

Find the second derivative of the function y = x/(x+1).

Okay, lets begin

First derivative: dy/dx = 1/(x+1)².

Now differentiate again to find the second derivative: d²y/dx² = d/dx [1/(x+1)²] = -2/(x+1)³

The second derivative is -2/(x+1)³.

Explanation

We first determine the first derivative, then apply the quotient rule to find the second derivative, simplifying as needed.

Well explained 👍

Problem 4

Prove: d/dx ((x/(x+1))³) = 3(x/(x+1))²/(x+1)²

Okay, lets begin

Let y = (x/(x+1))³. Using the chain rule: dy/dx = 3(x/(x+1))²·d/dx(x/(x+1)) d/dx(x/(x+1)) = 1/(x+1)².

Thus, dy/dx = 3(x/(x+1))²·1/(x+1)² = 3(x/(x+1))²/(x+1)² Hence proved.

Explanation

We apply the chain rule to differentiate the cubic function, ensuring to multiply by the derivative of the inner function.

Well explained 👍

Problem 5

Solve: d/dx (x/(x+1) + 2)

Okay, lets begin

Differentiate each term: d/dx(x/(x+1)) + d/dx(2) = 1/(x+1)² + 0 = 1/(x+1)²

Therefore, d/dx(x/(x+1) + 2) = 1/(x+1)²

Explanation

We differentiate each term separately and combine the results for the final derivative.

Well explained 👍

FAQs on the Derivative of x/(x+1)

1.What is the derivative of x/(x+1)?

Using the quotient rule for x/(x+1) gives: d/dx(x/(x+1)) = 1/(x+1)²

2.Can the derivative of x/(x+1) be used practically?

Yes, this derivative can be applied in real-world scenarios like calculating rates of change in economics and physics.

3.Is the derivative defined when x = -1?

No, at x = -1, the function is undefined, making it impossible to find a derivative.

4.What rule is used to differentiate x/(x+1)?

The quotient rule is used to differentiate x/(x+1), resulting in 1/(x+1)².

5.Is the derivative of x/(x+1) the same as that of (x+1)/x?

No, they are different functions and thus have different derivatives.

6.Can we find the derivative formula of x/(x+1)?

Yes, by applying the quotient rule to x/(x+1) and simplifying, we find the derivative formula: 1/(x+1)².

Important Glossaries for the Derivative of x/(x+1)

  • Derivative: Indicates how a function changes with respect to changes in its input.
  • Quotient Rule: A rule for differentiating functions expressed as a ratio of two differentiable functions.
  • Chain Rule: A rule for finding the derivative of composite functions.
  • Undefined: Points where a function does not exist, often leading to division by zero.
  • Simplification: The process of reducing expressions to their simplest form for clarity and ease of use.

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