Derivative of -1/x
2026-02-28 06:09 Diff
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Last updated on September 9, 2025

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.

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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:

Problem 1

Calculate the derivative of (-1/x)·(x²).

Okay, lets begin

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.

Well explained 👍

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.

Okay, lets begin

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.

Well explained 👍

Problem 3

Derive the second derivative of the function y = -1/x.

Okay, lets begin

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.

Well explained 👍

Problem 4

Prove: d/dx ((-1/x)²) = 2/x³.

Okay, lets begin

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.

Well explained 👍

Problem 5

Solve: d/dx (-1/x²).

Okay, lets begin

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.

Well explained 👍

FAQs on the Derivative of -1/x

1.Find the derivative of -1/x.

Using the power rule for -x⁻¹ gives, d/dx (-1/x) = 1/x².

2.Can we use the derivative of -1/x in real life?

Yes, we can use the derivative of -1/x in real life in calculating rates of change in various contexts, such as physics and engineering.

3.Is it possible to take the derivative of -1/x at the point where x = 0?

No, x = 0 is a point where -1/x is undefined, so it is impossible to take the derivative at this point (since the function does not exist there).

4.What rule is used to differentiate -1/x²?

We use the power rule to differentiate -1/x², d/dx (-1/x²) = 2/x³.

5.Are the derivatives of -1/x and -x the same?

No, they are different. The derivative of -1/x is 1/x², while the derivative of -x is -1.

6.Can we find the derivative of the -1/x formula?

To find, consider y = -1/x. We use the power rule: y’ = d/dx (-x⁻¹) = 1/x².

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

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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