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110 LearnersLast updated on September 9, 2025
Derivative of -1/x
We use the derivative of -1/x, which is 1/x², as a tool for understanding how this function changes in response to a slight change in x. Derivatives help us calculate profit or loss in real-life situations. We will now discuss the derivative of -1/x in detail.
What is the Derivative of -1/x?
We now understand the derivative of -1/x.
It is commonly represented as d/dx (-1/x) or (-1/x)', and its value is 1/x².
The function -1/x has a clearly defined derivative, indicating it is differentiable within its domain.
The key concepts are mentioned below:
Rational Function: (-1/x) is an example of a rational function.
Negative Exponent Rule: Used in simplifying derivatives of functions like -1/x.
Power Rule: A rule for differentiating expressions like x⁻¹.
Derivative of -1/x Formula
The derivative of -1/x can be denoted as d/dx (-1/x) or (-1/x)'. The formula we use to differentiate -1/x is: d/dx (-1/x) = 1/x² The formula applies to all x where x ≠ 0.
Proofs of the Derivative of -1/x
We can derive the derivative of -1/x using proofs.
To show this, we will use the differentiation rules.
There are several methods we use to prove this, such as:
Using the Power Rule
Using the Quotient Rule
We will now demonstrate that the differentiation of -1/x results in 1/x² using the above-mentioned methods:
Using the Power Rule
Consider the function f(x) = -1/x = -x⁻¹.
The power rule states that d/dx (xⁿ) = nxⁿ⁻¹.
Applying the power rule, we have: f'(x) = d/dx (-x⁻¹) = -(-1)x⁻² = 1/x².
Hence, proved.
Using the Quotient Rule
To prove the differentiation of -1/x using the quotient rule, We use the formula: d/dx (u/v) = (v·u' - u·v')/v².
Let u = -1 and v = x, then u' = 0 and v' = 1.
Applying the quotient rule: d/dx (-1/x) = (x·0 - (-1)·1)/x² = 1/x².
Therefore, the derivative of -1/x is 1/x².
Higher-Order Derivatives of -1/x
When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a little tricky.
To understand them better, think of a car where the speed changes (first derivative) and the rate at which the speed changes (second derivative) also changes.
Higher-order derivatives make it easier to understand functions like -1/x.
For the first derivative of a function, we write f′(x), which indicates how the function changes or its slope at a certain point.
The second derivative is derived from the first derivative, which is denoted using f′′(x).
Similarly, the third derivative, f′′′(x), is the result of the second derivative, and this pattern continues.
For the nth Derivative of -1/x, we generally use fⁿ(x) for the nth derivative of a function f(x), which tells us the change in the rate of change.
Special Cases:
When x = 0, the derivative is undefined because -1/x is undefined there. When x = 1, the derivative of -1/x = 1/(1²) = 1.
Common Mistakes and How to Avoid Them in Derivatives of -1/x
Students frequently make mistakes when differentiating -1/x. These mistakes can be resolved by understanding the proper solutions. Here are a few common mistakes and ways to solve them:
Mistake 1
Not simplifying the equation
Students may forget to simplify the equation, which can lead to incomplete or incorrect results. They often skip steps and directly arrive at the result, especially when solving using the power or quotient rule. Ensure that each step is written in order. Students might think it is awkward, but it is important to avoid errors in the process.
Mistake 2
Forgetting the Undefined Points of -1/x
They might not remember that -1/x is undefined at the point x = 0. Keep in mind that you should consider the domain of the function that you differentiate. It will help you understand that the function is not continuous at such certain points.
Mistake 3
Incorrect use of Quotient Rule
While differentiating functions such as -1/x, students misapply the quotient rule.
For example: Incorrect differentiation: d/dx (-1/x) = -1/x². d/dx (u/v) = (v·u' - u·v')/v² (where u = -1 and v = x). Applying the quotient rule, d/dx (-1/x) = (x·0 - (-1)·1)/x² = 1/x².
To avoid this mistake, write the quotient rule without errors. Always check for errors in the calculation and ensure it is properly simplified.
Mistake 4
Not writing Constants and Coefficients
There is a common mistake that students at times forget to multiply the constants placed before -1/x.
For example, they incorrectly write d/dx (5(-1/x)) = 1/x².
Students should check the constants in the terms and ensure they are multiplied properly.
For example, the correct equation is d/dx (5(-1/x)) = 5/x².
Mistake 5
Not Applying the Power Rule
Students often forget to use the power rule. This happens when the exponent of x is not considered.
For example: Incorrect: d/dx (-x⁻¹) = -1/x.
To fix this error, students should apply the power rule correctly, which gives d/dx (-x⁻¹) = 1/x².
Examples Using the Derivative of -1/x
Problem 1
Calculate the derivative of (-1/x)·(x²).
Here, we have f(x) = (-1/x)·x².
Using the product rule, f'(x) = u′v + uv′.
In the given equation, u = -1/x and v = x².
Let’s differentiate each term: u′ = d/dx (-1/x) = 1/x² v′ = d/dx (x²) = 2x
Substituting into the given equation, f'(x) = (1/x²)·(x²) + (-1/x)·(2x)
Let’s simplify terms to get the final answer, f'(x) = 1 - 2 = -1.
Thus, the derivative of the specified function is -1.
Explanation
We find the derivative of the given function by dividing the function into two parts. The first step is finding its derivative and then combining them using the product rule to get the final result.
Problem 2
A water tank is draining at a rate represented by the function y = -1/x, where y represents the rate of water level change at time x. If x = 2 hours, measure the rate of change of the water level.
We have y = -1/x (rate of change of water level) …(1)
Now, we will differentiate the equation (1)
Take the derivative of -1/x: dy/dx = 1/x²
Given x = 2 (substitute this into the derivative) dy/dx = 1/2² = 1/4
Hence, we get the rate of change of the water level at time x = 2 hours as 1/4.
Explanation
We find the rate of change of the water level at x = 2 hours as 1/4, which means that at a given point, the water level decreases at a rate of 1/4 the square of the time.
Problem 3
Derive the second derivative of the function y = -1/x.
The first step is to find the first derivative, dy/dx = 1/x² …(1)
Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [1/x²]
Here we use the power rule, d²y/dx² = -2/x³
Therefore, the second derivative of the function y = -1/x is -2/x³.
Explanation
We use the step-by-step process, where we start with the first derivative. Using the power rule, we differentiate 1/x². We then simplify the terms to find the final answer.
Problem 4
Prove: d/dx ((-1/x)²) = 2/x³.
Let’s start using the chain rule: Consider y = (-1/x)² = (x⁻¹)²
To differentiate, we use the chain rule: dy/dx = 2(x⁻¹)·d/dx (x⁻¹)
Since the derivative of x⁻¹ is -1/x², dy/dx = 2(x⁻¹)(-1/x²) = 2/x³
Hence proved.
Explanation
In this step-by-step process, we used the chain rule to differentiate the equation. Then, we replace x⁻¹ with its derivative. As a final step, we simplify to derive the equation.
Problem 5
Solve: d/dx (-1/x²).
To differentiate the function, we use the power rule:
Consider y = -1/x² = -x⁻²
Applying the power rule: dy/dx = 2x⁻³ = 2/x³
Therefore, d/dx (-1/x²) = 2/x³.
Explanation
In this process, we differentiate the given function using the power rule. As a final step, we simplify the equation to obtain the final result.
FAQs on the Derivative of -1/x
1.Find the derivative of -1/x.
2.Can we use the derivative of -1/x in real life?
3.Is it possible to take the derivative of -1/x at the point where x = 0?
4.What rule is used to differentiate -1/x²?
5.Are the derivatives of -1/x and -x the same?
6.Can we find the derivative of the -1/x formula?
Important Glossaries for the Derivative of -1/x
- Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.
- Power Rule: A fundamental rule of calculus used to find the derivative of functions of the form xⁿ.
- Quotient Rule: A rule used for differentiating functions that are divisions of two other functions.
- Undefined Points: Points where a function does not exist or is not continuous.
- Rational Function: A function that is the ratio of two polynomials, such as -1/x.
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Jaskaran Singh Saluja
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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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