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128 LearnersLast updated on December 15, 2025
Derivative of 1/(1-x)²
We use the derivative of 1/(1-x)² to understand how this function changes in response to small changes in x. Derivatives are crucial in many fields, such as physics and economics, for determining rates of change. We will now discuss the derivative of 1/(1-x)² in detail.
What is the Derivative of 1/(1-x)²?
The derivative of the function 1/(1-x)² is found using basic differentiation techniques.
It is represented as d/dx (1/(1-x)²) or (1/(1-x)²)'.
The function is differentiable within its domain (x ≠ 1).
The key concepts include:
Power Rule: A basic rule for differentiating functions of the form x^n.
Chain Rule: A rule used when differentiating a composition of functions.
Negative Exponent: Expressing the function as (1-x)^-2 for differentiation.
Proofs of the Derivative of 1/(1-x)²
We can derive the derivative of 1/(1-x)² using proofs.
We'll use the chain rule, considering the function as a composite function.
Here are the methods: Using the Chain Rule.
To prove the differentiation of 1/(1-x)² using the chain rule: Express the function as (1-x)^-2.
Consider u(x) = (1-x), then f(u) = u^-2.
By the chain rule: d/dx [f(u(x))] = f'(u) * u'(x).
Differentiate f(u) = u^-2: f'(u) = -2u^-3.
Differentiate u(x) = (1-x): u'(x) = -1.
Substitute these into the chain rule: d/dx [(1-x)^-2] = -2(1-x)^-3 * (-1) = 2/(1-x)³.
Hence, the derivative is 2/(1-x)³.
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Higher-Order Derivatives of 1/(1-x)²
When a function is differentiated several times, the resulting derivatives are called higher-order derivatives.
Higher-order derivatives can be more complex to handle.
For example, the second derivative of 1/(1-x)² is concerned with the rate of change of the rate of change.
For the first derivative, we have f′(x), which shows the slope or rate of change of the function.
The second derivative is derived from the first derivative, indicated by f′′(x).
In general, for the nth derivative of a function f(x), we use f n(x) to denote the nth derivative, which helps in analyzing the behavior of the function further.
Special Cases
When x approaches 1, the derivative becomes undefined because 1/(1-x)² has a vertical asymptote there.
When x is 0, the derivative of 1/(1-x)² is 2, since 1/(1-0)² = 1.
Common Mistakes and How to Avoid Them in Derivatives of 1/(1-x)²
Students often make mistakes when differentiating 1/(1-x)².
Understanding the correct methods can help avoid these errors.
Here are a few common mistakes and solutions:
Mistake 1
Incorrect Use of Negative Exponents
Students may forget to rewrite the function using negative exponents, leading to errors in applying the chain rule.
Remember to express 1/(1-x)² as (1-x)^-2 before differentiating to simplify the process.
Mistake 2
Ignoring Undefined Points
Students might not remember that 1/(1-x)² is undefined when x = 1.
Always consider the domain of the function when differentiating, as it will help in understanding the function's continuity.
Mistake 3
Misapplying the Chain Rule
When differentiating (1-x)^-2, students might misapply the chain rule.
For example, omitting to multiply by the derivative of the inner function, -1.
The correct application is crucial to avoid mistakes.
Mistake 4
Forgetting to Multiply by the Inner Derivative
A common mistake is neglecting to multiply by the derivative of the inner function (1-x).
Ensure to apply the chain rule fully by multiplying by -1, the derivative of (1-x).
Mistake 5
Simplification Errors
Students sometimes make errors in simplifying the final derivative.
Carefully simplify the expressions to avoid miscalculations.
Examples Using the Derivative of 1/(1-x)²
Problem 1
Calculate the derivative of (1/(1-x)²)·(1-x).
Here, we have f(x) = (1/(1-x)²)·(1-x).
Using the product rule, f'(x) = u′v + uv′.
In the given equation, u = 1/(1-x)² and v = (1-x).
Differentiate each term: u′ = d/dx (1/(1-x)²) = 2/(1-x)³.
v′ = d/dx (1-x) = -1.
Substituting into the product rule: f'(x) = (2/(1-x)³)·(1-x) + (1/(1-x)²)·(-1).
Simplify to get: f'(x) = 2/(1-x)² - 1/(1-x)² = 1/(1-x)².
Thus, the derivative of the specified function is 1/(1-x)².
Explanation
We find the derivative of the given function by expressing it in two parts.
We differentiate each part and then use the product rule to combine them for the final result.
Problem 2
A company is modeling a process where the rate of change is given by y = 1/(1-x)². If x = 0.5, find the rate of change.
We have y = 1/(1-x)² (rate of change)...(1) Differentiate equation (1) dy/dx = 2/(1-x)³.
Substitute x = 0.5 into the derivative: dy/dx = 2/(1-0.5)³ = 2/0.5³ = 2/(0.125) = 16.
Hence, the rate of change at x = 0.5 is 16.
Explanation
By substituting x = 0.5 into the derivative, we determine the rate of change of the process at that specific point, showing how sensitive the process is around that value.
Problem 3
Derive the second derivative of the function y = 1/(1-x)².
First, find the first derivative: dy/dx = 2/(1-x)³...(1) Differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [2/(1-x)³] Apply the chain rule: d²y/dx² = -6/(1-x)⁴ * (-1) = 6/(1-x)⁴.
Therefore, the second derivative of the function y = 1/(1-x)² is 6/(1-x)⁴.
Explanation
We start by finding the first derivative and apply the chain rule again to find the second derivative, showing the acceleration or curvature of the function.
Problem 4
Prove: d/dx ((1-x)⁻³) = 3(1-x)⁻⁴.
Start using the chain rule: Consider y = (1-x)⁻³.
Differentiate using the chain rule: dy/dx = -3(1-x)⁻⁴ * (-1).
Simplify: dy/dx = 3(1-x)⁻⁴.
Hence, proved.
Explanation
We used the chain rule to differentiate the function, simplifying by multiplying by the derivative of the inner function and verifying the result.
Problem 5
Solve: d/dx (1/(1-x)² + x).
To differentiate the function, differentiate each term separately: d/dx (1/(1-x)²) = 2/(1-x)³. d/dx (x) = 1.
Combine the derivatives: d/dx (1/(1-x)² + x) = 2/(1-x)³ + 1.
Therefore, the derivative is 2/(1-x)³ + 1.
Explanation
We differentiate each term of the function individually and combine the results for the final derivative.
FAQs on the Derivative of 1/(1-x)²
1.Find the derivative of 1/(1-x)².
2.Can the derivative of 1/(1-x)² be used in real life?
3.Is it possible to take the derivative of 1/(1-x)² at x = 1?
4.What rule is used to differentiate (1-x)^-2?
5.Are the derivatives of 1/(1-x)² and (1-x)⁻² the same?
Important Glossaries for the Derivative of 1/(1-x)²
- Derivative: The derivative of a function measures how the function changes as its input changes.
- Chain Rule: A rule for differentiating compositions of functions.
- Power Rule: A basic rule used to find the derivative of x raised to a power.
- Negative Exponent: Expressing a function with a negative exponent for simplification in differentiation.
- Asymptote: A line that a graph approaches but never touches.
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.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
