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113 LearnersLast updated on September 9, 2025
Derivative of x/(x+1)
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.
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:
Mistake 1
Incorrect Application of Quotient Rule
Students may apply the quotient rule incorrectly, leading to errors in differentiation. Ensure to properly follow the quotient rule formula and simplify each step accurately.
Mistake 2
Forgetting Undefined Points
Remember that the function x/(x+1) is undefined when x+1=0. Always consider the domain to avoid errors related to undefined points.
Mistake 3
Not Simplifying the Result
After applying the quotient rule, students might leave the expression unsimplified. Simplification is crucial to reach the correct derivative.
Mistake 4
Ignoring Higher-Order Derivatives
Some students fail to compute higher-order derivatives correctly, especially if the first derivative is not simplified. Understanding the process and rules is essential for accuracy.
Mistake 5
Not Applying the Chain Rule Correctly
In cases where the chain rule is needed for higher-order derivatives, incorrect application can lead to errors. Ensure clear identification of inner and outer functions for proper differentiation.
Examples Using the Derivative of x/(x+1)
Problem 1
Calculate the derivative of (x/(x+1))²
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.
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.
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.
Problem 3
Find the second derivative of the function y = x/(x+1).
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.
Problem 4
Prove: d/dx ((x/(x+1))³) = 3(x/(x+1))²/(x+1)²
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.
Problem 5
Solve: d/dx (x/(x+1) + 2)
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.
FAQs on the Derivative of x/(x+1)
1.What is the derivative of x/(x+1)?
2.Can the derivative of x/(x+1) be used practically?
3.Is the derivative defined when x = -1?
4.What rule is used to differentiate x/(x+1)?
5.Is the derivative of x/(x+1) the same as that of (x+1)/x?
6.Can we find the derivative formula of x/(x+1)?
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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Jaskaran Singh Saluja
About the Author
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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