BrightChamps Logo
Login
Creative Math Ideas Image
Live Math Learners Count Icon102 Learners

Last updated on July 22nd, 2025

Math Whiteboard Illustration

Derivative of 1/x^7

Professor Greenline Explaining Math Concepts

We use the derivative of 1/x^7, which is -7/x^8, to understand how this function changes with a small change in x. Derivatives are useful in calculating rates of change, like speed or growth, in real-world applications. We will now discuss the derivative of 1/x^7 in detail.

Derivative of 1/x^7 for Vietnamese Students
Professor Greenline from BrightChamps

What is the Derivative of 1/x^7?

To find the derivative of 1/x^7, we represent it as d/dx (1/x^7) or (1/x^7)'. The derivative is found to be -7/x^8. This function is differentiable within its domain, indicating a smooth rate of change.

 

The key concepts are mentioned below:

 

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

 

Negative Exponents: 1/x^7 can be rewritten as x^-7.

 

Chain Rule: Useful for functions within functions.

Professor Greenline from BrightChamps

Derivative of 1/x^7 Formula

The derivative of 1/x^7 can be denoted as d/dx (1/x^7) or (1/x^7)'.

 

Using the power rule, the formula we use to differentiate 1/x^7 is: d/dx (1/x^7) = -7/x^8

 

This formula applies to all x where x ≠ 0.

Professor Greenline from BrightChamps

Proofs of the Derivative of 1/x^7

We can derive the derivative of 1/x^7 through several methods. Here we demonstrate using the power rule and chain rule:

 

By Power Rule

 

Rewrite 1/x^7 as x^-7. Using the power rule, d/dx (x^n) = n*x^(n-1), we find: d/dx (x^-7) = -7*x^(-7-1) = -7/x^8.

Hence, the derivative is -7/x^8.

 

Using Chain Rule

 

Consider f(x) = 1/x^7 = (x^7)^-1. Let g(x) = x^7, then f(x) = g(x)^-1.

 

Using the chain rule, d/dx (g(x)^n) = n*g(x)^(n-1)*g'(x), we get: d/dx (1/x^7) = -1*(x^7)^-2*(7x^6) = -7/x^8.

 

Thus, the derivative is -7/x^8.

Professor Greenline from BrightChamps

Higher-Order Derivatives of 1/x^7

When a function is differentiated multiple times, we obtain higher-order derivatives. Think of it like the acceleration of a car (second derivative) in addition to the speed (first derivative). Higher-order derivatives provide deeper insights into the behavior of functions like 1/x^7.

 

For the first derivative, we write f′(x), indicating the rate of change or slope at a point. The second derivative, f′′(x), is derived from the first derivative. This pattern continues for higher-order derivatives.

 

For the nth Derivative of 1/x^7, we denote it as f^(n)(x), showing changes in the rate of change.

Professor Greenline from BrightChamps

Special Cases:

When x is 0, the derivative is undefined because 1/x^7 has a vertical asymptote there. When x is 1, the derivative of 1/x^7 = -7.

Max Pointing Out Common Math Mistakes

Common Mistakes and How to Avoid Them in Derivatives of 1/x^7

Students often make errors when differentiating 1/x^7. Understanding the correct methods can help avoid these mistakes. Here are some common errors and solutions:

Mistake 1

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Ignoring Negative Exponents

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students sometimes overlook rewriting 1/x^7 as x^-7, leading to incorrect differentiation. Always rewrite the expression in terms of exponents before applying the power rule.

Mistake 2

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Forgetting Undefined Points

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

It's easy to forget that 1/x^7 is undefined at x = 0. Always consider the domain when differentiating functions to understand where the function isn't continuous.

Mistake 3

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Incorrect Use of the Power Rule

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Errors occur when students misapply the power rule. For example: Incorrect differentiation: d/dx (1/x^7) = 7/x^6. Correctly apply the power rule: x^-7, so the derivative is -7/x^8.

Mistake 4

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Misapplying the Chain Rule

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students sometimes misapply the chain rule when differentiating composite functions. Ensure that you identify the outer and inner functions correctly and differentiate each accurately.

Mistake 5

Red Cross Icon Indicating Mistakes to Avoid in This Math Topic

Missing Negative Signs

Green Checkmark Icon Indicating Correct Solutions in This Math Topic

Students may forget to include the negative sign in the derivative of x^-n. For instance, write -7 instead of 7 when differentiating x^-7.

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

Examples Using the Derivative of 1/x^7

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

Problem 1

Calculate the derivative of (1/x^7)·(x^2).

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

Here, we have f(x) = (1/x^7)·(x^2).

 

Using the product rule, f'(x) = u′v + uv′ In the given equation, u = 1/x^7 and v = x^2.

 

Differentiate each term: u′= d/dx (1/x^7) = -7/x^8 v′= d/dx (x^2) = 2x

 

Substitute into the product rule, f'(x) = (-7/x^8)·(x^2) + (1/x^7)·(2x)

 

Simplify to get the final answer, f'(x) = -7/x^6 + 2/x^6

 

Thus, the derivative of the specified function is -5/x^6.

Explanation

We find the derivative by dividing the function into two parts. First, we find their individual derivatives, then combine them using the product rule for the final result.

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

Problem 2

A company models a certain variable with the function y = 1/x^7. If x = 2, determine the rate of change.

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

We have y = 1/x^7 (the model of the variable)...(1)

 

Now, differentiate equation (1): dy/dx = -7/x^8

 

Given x = 2, substitute this into the derivative: dy/dx = -7/(2^8) dy/dx = -7/256

 

Hence, the rate of change when x = 2 is -7/256.

Explanation

We find the rate of change by substituting x = 2 into the derivative -7/x^8. This calculation shows the rate at which the variable changes at x = 2.

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 = 1/x^7.

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

First, find the first derivative: dy/dx = -7/x^8...(1)

 

Now, differentiate equation (1) to get the second derivative: d^2y/dx^2 = d/dx [-7/x^8]

 

Use the power rule: d^2y/dx^2 = 56/x^9

 

Therefore, the second derivative of the function y = 1/x^7 is 56/x^9.

Explanation

We use the step-by-step process, starting with the first derivative. Using the power rule again, we differentiate -7/x^8 to find the second derivative, 56/x^9.

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

Problem 4

Prove: d/dx (x^2/x^7) = -5/x^6.

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

Rewrite x^2/x^7 as x^-5.

 

Differentiate using the power rule: d/dx (x^-5) = -5*x^(-5-1) d/dx (x^-5) = -5/x^6

 

Hence proved.

Explanation

In this step-by-step process, we rewrite the function with a negative exponent, differentiate using the power rule, and simplify to derive the equation.

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

Problem 5

Solve: d/dx (x/x^7).

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

Rewrite x/x^7 as x^-6.

 

Differentiate using the power rule: d/dx (x^-6) = -6*x^(-6-1) d/dx (x^-6) = -6/x^7

 

Therefore, d/dx (x/x^7) = -6/x^7.

Explanation

In this process, we rewrite the given function with a negative exponent and differentiate using the power rule to simplify the equation and find the final result.

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

FAQs on the Derivative of 1/x^7

1.Find the derivative of 1/x^7.

Math FAQ Answers Dropdown Arrow

2.Can we use the derivative of 1/x^7 in real life?

Math FAQ Answers Dropdown Arrow

3.Is it possible to take the derivative of 1/x^7 at the point where x = 0?

Math FAQ Answers Dropdown Arrow

4.What rule is used to differentiate 1/x^7?

Math FAQ Answers Dropdown Arrow

5.Are the derivatives of 1/x^7 and (1/x)^7 the same?

Math FAQ Answers Dropdown Arrow
Professor Greenline from BrightChamps

Important Glossaries for the Derivative of 1/x^7

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

 

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

 

  • Negative Exponent: Represents the reciprocal of a base raised to a positive exponent, e.g., x^-n = 1/x^n.

 

  • Chain Rule: A rule used to differentiate composite functions.

 

  • Asymptote: A line that a graph approaches but never touches or crosses.
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