BrightChamps Logo
Login
Creative Math Ideas Image
Live Math Learners Count Icon105 Learners

Last updated on July 21st, 2025

Math Whiteboard Illustration

Derivative of 3/x

Professor Greenline Explaining Math Concepts

We use the derivative of 3/x, which is -3/x², as a tool to measure how the function changes in response to a slight change in x. Derivatives help us calculate profit or loss in real-life situations. We will now talk about the derivative of 3/x in detail.

Derivative of 3/x for Indian Students
Professor Greenline from BrightChamps

What is the Derivative of 3/x?

We now understand the derivative of 3/x. It is commonly represented as d/dx (3/x) or (3/x)', and its value is -3/x².

 

The function 3/x has a clearly defined derivative, indicating it is differentiable within its domain.

 

The key concepts are mentioned below: Reciprocal Function: (f(x) = 3/x).

 

Power Rule: A rule for differentiating functions of the form x^n.

 

Negative Exponent: 3/x can be written as 3x^(-1).

Professor Greenline from BrightChamps

Derivative of 3/x Formula

The derivative of 3/x can be denoted as d/dx (3/x) or (3/x)'.

 

The formula we use to differentiate 3/x is: d/dx (3/x) = -3/x² The formula applies to all x where x ≠ 0.

Professor Greenline from BrightChamps

Proofs of the Derivative of 3/x

We can derive the derivative of 3/x using proofs. To show this, we will use the rules of differentiation.

 

There are several methods we use to prove this, such as: By First Principle Using Power Rule Using Product Rule

 

We will now demonstrate that the differentiation of 3/x results in -3/x² using the above-mentioned methods: By First Principle The derivative of 3/x can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.

 

To find the derivative of 3/x using the first principle, we will consider f(x) = 3/x.

 

Its derivative can be expressed as the following limit. f'(x) = limₕ→₀ [f(x + h) - f(x)] / h … (1) Given that f(x) = 3/x, we write f(x + h) = 3/(x + h).

 

Substituting these into equation (1), f'(x) = limₕ→₀ [3/(x + h) - 3/x] / h = limₕ→₀ [3x - 3(x + h)] / [x(x + h)h] = limₕ→₀ [-3h] / [x(x + h)h] = limₕ→₀ -3 / [x(x + h)] = -3 / x² Hence, the derivative of 3/x is -3/x².

 

Using Power Rule To prove the differentiation of 3/x using the power rule, We express 3/x as 3x⁻¹.

 

Using the power rule, d/dx [x^n] = n*x^(n-1), d/dx [3x⁻¹] = -1*3x^(-1-1) = -3x⁻² = -3/x²

 

Thus, the derivative of 3/x is -3/x².

 

Using Product Rule We will now prove the derivative of 3/x using the product rule.

 

The step-by-step process is demonstrated below: Here, we use the formula, 3/x = 3 * x⁻¹ Let u = 3 and v = x⁻¹

 

Using the product rule formula: d/dx [u*v] = u'v + uv' u' = d/dx (3) = 0 v' = d/dx (x⁻¹) = -1*x⁻² d/dx (3/x) = 0 * x⁻¹ + 3 * (-1)x⁻² = -3x⁻² = -3/x²

 

Therefore, the derivative of 3/x is -3/x².

Professor Greenline from BrightChamps

Higher-Order Derivatives of 3/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 3/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 3/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 (continuing for higher-order derivatives).

Professor Greenline from BrightChamps

Special Cases:

When x = 0, the derivative is undefined because 3/x has a vertical asymptote there. When x = 1, the derivative of 3/x = -3/1², which is -3.

Max Pointing Out Common Math Mistakes

Common Mistakes and How to Avoid Them in Derivatives of 3/x

Students frequently make mistakes when differentiating 3/x.

 

These mistakes can be resolved by understanding the proper solutions. Here are a few common mistakes and ways to solve them:

Mistake 1

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Misapplying the power rule

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students may forget to apply the power rule correctly, especially when dealing with negative exponents.

 

They might incorrectly write the derivative of x⁻¹ as x⁰.

 

Ensure that the power rule is applied correctly by subtracting one from the exponent and multiplying by the original exponent.

Mistake 2

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Forgetting the Undefined Points

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

They might not remember that 3/x is undefined at 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

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Incorrect use of Product Rule

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

While differentiating expressions like 3/x, students misapply the product rule or fail to recognize it as a product of 3 and x⁻¹.

 

For example, incorrect differentiation: d/dx (3/x) = 3 * d/dx (x⁻¹). The correct application: d/dx (3/x) = 3 * (-1) * x⁻² = -3/x².

Mistake 4

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Not writing Constants and Coefficients

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

There is a common mistake where students sometimes forget to keep track of constants. For example, they might incorrectly write d/dx (3/x) as x⁻².

 

Students should ensure they multiply the constant correctly, leading to the correct equation: d/dx (3/x) = -3/x².

Mistake 5

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Not Applying the Power Rule

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students often forget to use the power rule when the function is in the form of x raised to a negative power.

 

For example, incorrect: d/dx (3x⁻¹) = 3x⁰. To fix this error, students should apply the power rule properly: d/dx (3x⁻¹) = -3x⁻².

arrow-right
Max from BrightChamps Saying "Hey"
Hey!

Examples Using the Derivative of 3/x

Ray, the Character from BrightChamps Explaining Math Concepts
Max, the Girl Character from BrightChamps

Problem 1

Calculate the derivative of (3/x * x³)

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

Here, we have f(x) = (3/x) * x³. Using the product rule, f'(x) = u′v + uv′ In the given equation, u = 3/x and v = x³. Let’s differentiate each term, u′ = d/dx (3/x) = -3/x² v′ = d/dx (x³) = 3x²

 

Substituting into the given equation, f'(x) = (-3/x²) * x³ + (3/x) * 3x²

 

Let’s simplify terms to get the final answer, f'(x) = -3x + 9 Thus, the derivative of the specified function is -3x + 9.

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.

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Max, the Girl Character from BrightChamps

Problem 2

A water tank has a drain that empties at a rate proportional to 3/x, where x is the time in minutes. If x = 2 minutes, calculate the change in the drainage rate.

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

We have y = 3/x (drainage rate)...(1) Now, we will differentiate the equation (1).

 

Take the derivative 3/x: dy/dx = -3/x² Given x = 2 (substitute this into the derivative) dy/dx = -3/2² = -3/4

 

Hence, the change in the drainage rate at x = 2 minutes is -3/4.

Explanation

We find the change in the drainage rate at x = 2 minutes as -3/4, which means that at this point, the rate of drainage decreases by 3/4 units per minute.

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Max, the Girl Character from BrightChamps

Problem 3

Derive the second derivative of the function y = 3/x.

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

The first step is to find the first derivative, dy/dx = -3/x²...(1)

 

Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [-3/x²]

 

Here we use the power rule, d²y/dx² = 6/x³ Therefore, the second derivative of the function y = 3/x is 6/x³.

Explanation

We use the step-by-step process, where we start with the first derivative.

 

Using the power rule, we differentiate -3/x². We then simplify the terms to find the final answer.

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Max, the Girl Character from BrightChamps

Problem 4

Prove: d/dx (3/x²) = -6/x³.

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

Let’s start using the power rule: Consider y = 3/x² y = 3x⁻² To differentiate, we use the power rule: dy/dx = -2 * 3 * x⁻³ = -6x⁻³ = -6/x³ Hence proved.

Explanation

In this step-by-step process, we used the power rule to differentiate the equation 3x⁻². Then, we simplify the expression to derive the equation.

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Max, the Girl Character from BrightChamps

Problem 5

Solve: d/dx (3/x + x)

Ray, the Boy Character from BrightChamps Saying "Let’s Begin"
Okay, lets begin

To differentiate the function, we use the sum rule: d/dx (3/x + x) = d/dx (3/x) + d/dx (x) We will substitute d/dx (3/x) = -3/x² and d/dx (x) = 1 = -3/x² + 1 Therefore, d/dx (3/x + x) = -3/x² + 1

Explanation

In this process, we differentiate the given function using the sum rule.

 

As a final step, we simplify the equation to obtain the final result.

Max from BrightChamps Praising Clear Math Explanations
Well explained 👍
Ray Thinking Deeply About Math Problems

FAQs on the Derivative of 3/x

1.Find the derivative of 3/x.

Math FAQ Answers Dropdown Arrow

2.Can we use the derivative of 3/x in real life?

Math FAQ Answers Dropdown Arrow

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

Math FAQ Answers Dropdown Arrow

4.What rule is used to differentiate 3/x * x³?

Math FAQ Answers Dropdown Arrow

5.Are the derivatives of 3/x and 3x⁻¹ the same?

Math FAQ Answers Dropdown Arrow
Professor Greenline from BrightChamps

Important Glossaries for the Derivative of 3/x

  • Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.

 

  • Reciprocal Function: A function of the form f(x) = 3/x.

 

  • Power Rule: A rule used to differentiate functions of the form x^n.

 

  • Undefined Points: Points at which the function has no value, such as x = 0 for 3/x.

 

  • Higher-Order Derivative: Derivatives taken multiple times to understand the rate of change of the rate of change of a function.
Math Teacher Background Image
Math Teacher Image

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.

Max, the Girl Character from BrightChamps

Fun Fact

: He loves to play the quiz with kids through algebra to make kids love it.

INDONESIA - Axa Tower 45th floor, JL prof. Dr Satrio Kav. 18, Kel. Karet Kuningan, Kec. Setiabudi, Kota Adm. Jakarta Selatan, Prov. DKI Jakarta
INDIA - H.No. 8-2-699/1, SyNo. 346, Rd No. 12, Banjara Hills, Hyderabad, Telangana - 500034
SINGAPORE - 60 Paya Lebar Road #05-16, Paya Lebar Square, Singapore (409051)
USA - 251, Little Falls Drive, Wilmington, Delaware 19808
VIETNAM (Office 1) - Hung Vuong Building, 670 Ba Thang Hai, ward 14, district 10, Ho Chi Minh City
VIETNAM (Office 2) - 143 Nguyễn Thị Thập, Khu đô thị Him Lam, Quận 7, Thành phố Hồ Chí Minh 700000, Vietnam
UAE - BrightChamps, 8W building 5th Floor, DAFZ, Dubai, United Arab Emirates
UK - Ground floor, Redwood House, Brotherswood Court, Almondsbury Business Park, Bristol, BS32 4QW, United Kingdom