Derivative of 6/x
2026-02-28 06:11 Diff
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Last updated on September 27, 2025

We use the derivative of 6/x, which is -6/x², as a tool to understand how the function changes with respect to x. Derivatives help us analyze various real-world scenarios involving rates of change. We will now discuss the derivative of 6/x in detail.

What is the Derivative of 6/x?

We now understand the derivative of 6/x. It is commonly represented as d/dx (6/x) or (6/x)', and its value is -6/x². The function 6/x has a clearly defined derivative, indicating it is differentiable within its domain.

The key concepts are mentioned below: 

  • Function Representation: (6/x is equivalent to 6·x⁻¹). 
     
  • Power Rule: Rule for differentiating functions of the form axⁿ. 
     
  • Negative Exponents: Understanding that x⁻¹ is equivalent to 1/x.

Derivative of 6/x Formula

The derivative of 6/x can be denoted as d/dx (6/x) or (6/x)'.The formula we use to differentiate 6/x is: d/dx (6/x) = -6/x² (or) (6/x)' = -6/x²

The formula applies to all x where x ≠ 0.

Proofs of the Derivative of 6/x

We can derive the derivative of 6/x using proofs. To show this, we will use algebraic manipulation along with the rules of differentiation.

There are several methods we use to prove this, such as:

  • By First Principle
     
  • Using Power Rule

By First Principle

The derivative of 6/x can be proven using the First Principle, which expresses the derivative as the limit of the difference quotient. To find the derivative of 6/x using the first principle, we will consider f(x) = 6/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) = 6/x, we write f(x + h) = 6/(x + h). Substituting these into equation (1), f'(x) = limₕ→₀ [6/(x + h) - 6/x] / h = limₕ→₀ [(6x - 6(x + h)) / (x(x + h))] / h = limₕ→₀ [-6h / (x² + xh)] / h = limₕ→₀ [-6 / (x² + xh)] As h approaches 0, the expression becomes: f'(x) = -6/x² Hence, proved.

Using Power Rule

To prove the differentiation of 6/x using the power rule, We rewrite the function as 6·x⁻¹ and apply the power rule: d/dx [6·x⁻¹] = 6·(-1)x⁻² = -6/x² Thus, the derivative of 6/x is -6/x².

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Higher-Order Derivatives of 6/x

When a function is differentiated several times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a bit complex. To understand them better, consider a scenario where certain quantities change at different rates. Higher-order derivatives help analyze such functions effectively.

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 6/x, we generally use fⁿ(x) for the nth derivative of a function f(x) to understand the rate of change at different levels.

Special Cases:

When x = 0, the derivative is undefined because 6/x has a vertical asymptote there.

When x = 1, the derivative of 6/x = -6/(1)², which is -6.

Common Mistakes and How to Avoid Them in Derivatives of 6/x

Students frequently make mistakes when differentiating 6/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 (6/x)·(x²)

Okay, lets begin

Here, we have f(x) = (6/x)·(x²). Using the product rule, f'(x) = u′v + uv′ In the given equation, u = 6/x and v = x². Let’s differentiate each term, u′ = d/dx (6/x) = -6/x² v′ = d/dx (x²) = 2x Substituting into the given equation, f'(x) = (-6/x²)·(x²) + (6/x)·(2x) Simplifying terms, f'(x) = -6 + 12 = 6 Thus, the derivative of the specified function is 6.

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 company finds that the cost of production is represented by the function y = 6/x where y represents the cost at production level x. If x = 3 units, calculate the rate of change of cost.

Okay, lets begin

We have y = 6/x (cost function)...(1) Now, we will differentiate the equation (1). Take the derivative of 6/x: dy/dx = -6/x² Given x = 3, substitute this into the derivative: dy/dx = -6/(3)² = -6/9 = -2/3 Hence, the rate of change of cost at a production level of x = 3 units is -2/3.

Explanation

We find the rate of change of cost at x = 3 units as -2/3, indicating that for every unit increase in production, the cost decreases by 2/3 units.

Well explained 👍

Problem 3

Derive the second derivative of the function y = 6/x.

Okay, lets begin

The first step is to find the first derivative, dy/dx = -6/x²...(1) Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [-6/x²] Using the power rule, d²y/dx² = 12/x³ Therefore, the second derivative of the function y = 6/x is 12/x³.

Explanation

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

Using the power rule, we differentiate -6/x².

We then simplify the terms to find the final answer.

Well explained 👍

Problem 4

Prove: d/dx (3/x²) = -6/x³.

Okay, lets begin

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

Explanation

In this step-by-step process, we used the power rule to differentiate the equation.

Then, we replace x⁻² with its derivative to derive the equation.

Well explained 👍

Problem 5

Solve: d/dx (6/x²)

Okay, lets begin

To differentiate the function, we rewrite it using exponents: d/dx (6/x²) = d/dx (6·x⁻²) Applying the power rule, = 6·(-2)x⁻³ = -12/x³ Therefore, d/dx (6/x²) = -12/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 6/x

1.Find the derivative of 6/x.

Using the power rule for 6·x⁻¹ gives, d/dx (6/x) = -6/x² (simplified).

2.Can we use the derivative of 6/x in real life?

Yes, we can use the derivative of 6/x in real life in areas such as economics to determine the rate of change of cost with respect to production levels.

3.Is it possible to take the derivative of 6/x at the point where x = 0?

No, x = 0 is a point where 6/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 6/x²?

We use the power rule to differentiate 6/x² by rewriting it as 6·x⁻², d/dx (6·x⁻²) = -12/x³.

5.Are the derivatives of 6/x and 6x⁻¹ the same?

Yes, they are the same. The derivative of 6/x is -6/x², which is equivalent to differentiating 6x⁻¹ using the power rule.

Important Glossaries for the Derivative of 6/x

  • Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. 
  • Power Rule: A basic differentiation rule used to find the derivative of a function of the form axⁿ. 
  • Undefined Points: Points where a function does not have a value, often due to division by zero.
  • Rate of Change: A measure of how a quantity changes with respect to another variable. 
  • Asymptote: A line that a curve approaches but never touches or intersects.

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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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