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243 LearnersLast updated on October 22, 2025
Derivative of -1/x²
What is the Derivative of -1/x²?
We now understand the derivative of -1/x². It is commonly represented as d/dx (-1/x²) or (-1/x²)', and its value is 2/x³. The function -1/x² has a clearly defined derivative, indicating it is differentiable within its domain.
The key concepts are mentioned below:
Function: -1/x² is a simple rational function.
Power Rule: Rule for differentiating functions of the form x^n.
Negative Exponents: Understanding how to handle derivatives of negative powers.
Derivative of -1/x² Formula
The derivative of -1/x² can be denoted as d/dx (-1/x²) or (-1/x²)'.
The formula we use to differentiate -1/x² is: d/dx (-1/x²) = 2/x³
The formula applies to all x where x ≠ 0.
Proofs of the Derivative of -1/x²
We can derive the derivative of -1/x² using proofs. To show this, we will use differentiation rules. There are several methods we use to prove this, such as:
Using Power Rule
Using Chain Rule
We will now demonstrate that the differentiation of -1/x² results in 2/x³ using the above-mentioned methods:
Using Power Rule The derivative of -1/x² can be found using the Power Rule, which expresses the derivative of x^n as n*x^(n-1).
To find the derivative of -1/x², rewrite the function as f(x) = -x⁻².
Its derivative can be expressed as: f'(x) = d/dx (-x⁻²) = (-2)x^(-2-1) = 2/x³. Thus, the derivative is 2/x³. Hence, proved.
Using Chain Rule
To prove the differentiation of -1/x² using the chain rule, Consider y = -1/x² = -(x⁻²).
Using the chain rule: d/dx [u(x)^n] = n*u(x)^(n-1)*u'(x) Let u(x) = x and n = -2. dy/dx = -2*x^(-2-1). dy/dx = 2/x³.
Thus, the derivative of -1/x² is 2/x³.
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Higher-Order Derivatives of -1/x²
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/x². 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/x², we generally use fⁿ(x) for the nth derivative of a function f(x), which tells us the change in the rate of change (continuing for higher-order derivatives).
Special Cases:
When x is 0, the derivative is undefined because -1/x² has a vertical asymptote there.
When x is 1, the derivative of -1/x² = 2/1³, which is 2.
Common Mistakes and How to Avoid Them in Derivatives of -1/x²
Students frequently make mistakes when differentiating -1/x². 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 applying the power or chain 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/x²
They might not remember that -1/x² 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 Power Rule
While differentiating functions like -1/x², students misapply the power rule.
For example: Incorrect differentiation: d/dx (-x⁻²) = -2x⁻².
Correct use of power rule: d/dx (-x⁻²) = -2*x^(-2-1) = 2/x³. To avoid this mistake, write the power 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 x terms.
For example, they incorrectly write d/dx (-5/x²) = -2/x³. Students should check the constants in the terms and ensure they are multiplied properly.
For example, the correct equation is d/dx (-5/x²) = 10/x³.
Mistake 5
Not Applying the Chain Rule
Students often forget to use the chain rule. This happens when the derivative of the inner function is not considered.
For example: Incorrect: d/dx (-(2x)⁻²) = 2(2x)⁻³. To fix this error, students should divide the functions into inner and outer parts. Then, make sure that each function is differentiated. For example, d/dx (-(2x)⁻²) = 8/x³.
Examples Using the Derivative of -1/x²
Problem 1
Calculate the derivative of (-1/x²)·(x³)
Here, we have f(x) = (-1/x²)·(x³). Using the product rule, f'(x) = u′v + uv′
In the given equation, u = -1/x² and v = x³.
Let’s differentiate each term, u′= d/dx (-1/x²) = 2/x³ v′= d/dx (x³) = 3x²
Substituting into the given equation, f'(x) = (2/x³)·(x³) + (-1/x²)·(3x²)
Let’s simplify terms to get the final answer, f'(x) = 2 - 3 = -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 designed as a paraboloid, and the rate of change of the water level with respect to time is represented by the function y = -1/x², where y is the water level at time x. If x = 2 hours, measure the rate of change of the water level.
We have y = -1/x² (rate of change of water level)...(1)
Now, we will differentiate the equation (1)
Take the derivative -1/x²: dy/dx = 2/x³
Given x = 2 (substitute this into the derivative) dy/dx = 2/2³ = 2/8 = 0.25
Hence, we get the rate of change of the water level at x = 2 as 0.25.
Explanation
We find the rate of change of the water level at x = 2 as 0.25, which means that at a given point, the water level decreases at a rate of 0.25 per unit time.
Problem 3
Derive the second derivative of the function y = -1/x².
The first step is to find the first derivative, dy/dx = 2/x³...(1)
Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [2/x³]
Here we use the power rule, d²y/dx² = -6/x⁴
Therefore, the second derivative of the function y = -1/x² is -6/x⁴.
Explanation
We use the step-by-step process, where we start with the first derivative. Using the power rule, we differentiate 2/x³. We then simplify the terms to find the final answer.
Problem 4
Prove: d/dx ((-1/x²)²) = 4/x⁵.
Let’s start using the chain rule: Consider y = (-1/x²)² = [(-1)²] * (x⁻²)²
To differentiate, we use the chain rule: dy/dx = 2 * (-1/x²) * d/dx (-1/x²)
Since the derivative of -1/x² is 2/x³, dy/dx = 4/x⁵ Hence proved.
Explanation
In this step-by-step process, we used the chain rule to differentiate the equation. Then, we replace -1/x² with its derivative. As a final step, we simplify the expression to derive the equation.
Problem 5
Solve: d/dx (-1/x² + x)
To differentiate the function, d/dx (-1/x² + x) = d/dx (-1/x²) + d/dx (x)
We will substitute d/dx (-1/x²) = 2/x³ and d/dx (x) = 1 = 2/x³ + 1
Therefore, d/dx (-1/x² + x) = 2/x³ + 1
Explanation
In this process, we differentiate the given function term by term. As a final step, we simplify the equation to obtain the final result.
FAQs on the Derivative of -1/x²
1.Find the derivative of -1/x².
2.Can we use the derivative of -1/x² in real life?
3.Is it possible to take the derivative of -1/x² at the point where x = 0?
4.What rule is used to differentiate -1/x²?
5.Are the derivatives of -1/x² and (-x²)⁻¹ the same?
Important Glossaries for the Derivative of -1/x²
- Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.
- Rational Function: A function that can be expressed as the ratio of two polynomials.
- Power Rule: A basic rule for differentiating functions of the form xn.
- Chain Rule: A rule for differentiating compositions of functions.
- Asymptote: A line that a graph approaches but never touches.
Vikrant Sharma
About the Author
Vikrant is a passionate Mathematics teacher with over 5 years of teaching experience in both conventional and Vedic Maths. His student-focused approach combines clear concepts with motivation and encouragement, helping students build confidence and develo
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
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