Derivative of 1/x^7
2026-02-28 23:50 Diff
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Last updated on August 5, 2025

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.

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.

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.

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.

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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.

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.

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:

Problem 1

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

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.

Well explained 👍

Problem 2

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

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.

Well explained 👍

Problem 3

Derive the second derivative of the function y = 1/x^7.

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.

Well explained 👍

Problem 4

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

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.

Well explained 👍

Problem 5

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

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.

Well explained 👍

FAQs on the Derivative of 1/x^7

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

Rewrite using negative exponents: 1/x^7 = x^-7. Differentiate using the power rule: d/dx (x^-7) = -7/x^8.

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

Yes, the derivative of 1/x^7 can be used to model various real-life phenomena involving rates of change, such as decay processes or inverse relationships, in fields like physics or engineering.

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

No, x = 0 is a point where 1/x^7 is undefined, so the derivative cannot be taken there (since the function does not exist at x = 0).

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

The power rule is used to differentiate 1/x^7, rewritten as x^-7. The derivative is -7/x^8.

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

No, they are different. The derivative of 1/x^7 is -7/x^8, while the derivative of (1/x)^7, which is x^-7, is also -7/x^8, but the context and interpretation of the function may differ.

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.

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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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