105 LearnersLast updated on July 21st, 2025
Derivative of 4/x
We use the derivative of 4/x, which is -4/x², as a measuring tool for 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 discuss the derivative of 4/x in detail.
What is the Derivative of 4/x?
We now understand the derivative of 4/x.
It is commonly represented as d/dx (4/x) or (4/x)', and its value is -4/x².
The function 4/x has a clearly defined derivative, indicating it is differentiable within its domain.
The key concepts are mentioned below: Reciprocal Function: (4/x is a constant divided by x).
Power Rule: Rule for differentiating x raised to a power. Quotient Rule: Used for differentiating expressions like 4/x.
Derivative of 4/x Formula
The derivative of 4/x can be denoted as d/dx (4/x) or (4/x)'.
The formula we use to differentiate 4/x is: d/dx (4/x) = -4/x² The formula applies to all x ≠ 0.
Proofs of the Derivative of 4/x
We can derive the derivative of 4/x using proofs. To show this, we will use basic 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 4/x results in -4/x² using the above-mentioned methods: Using the Power Rule The derivative of 4/x can be expressed by rewriting it as 4x⁻¹.
To find the derivative, consider f(x) = 4x⁻¹. Using the power rule, which states that d/dx (xⁿ) = n·xⁿ⁻¹, f'(x) = 4(-1)x⁻² f'(x) = -4/x²
Hence, proved. Using the Quotient Rule To prove the differentiation of 4/x using the quotient rule, We use the formula: d/dx (u/v) = (v·u' - u·v') / v² Let u = 4 and v = x, u' = d/dx (4) = 0 v' = d/dx (x) = 1
Substituting these into the quotient rule, d/dx (4/x) = (x·0 - 4·1) / x² d/dx (4/x) = -4/x² Hence, proved.
Higher-Order Derivatives of 4/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 4/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 4/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).
Special Cases:
When x is 0, the derivative is undefined because the function 4/x is undefined at x = 0.
As x approaches infinity, the derivative of 4/x approaches 0, indicating the slope becomes flatter.
Common Mistakes and How to Avoid Them in Derivatives of 4/x
Students frequently make mistakes when differentiating 4/x. These mistakes can be resolved by understanding the proper solutions.
Here are a few common mistakes and ways to solve them:
Mistake 1
Not Simplifying the Equation
Students may forget to simplify the equation, which can lead to incomplete or incorrect results.
They often skip steps and directly arrive at the result, especially when solving using the quotient rule.
Ensure that each step is written in order. Students might think it is awkward, but it is important to avoid errors in the process.
Mistake 2
Forgetting the Undefined Points of 4/x
They might not remember that 4/x is undefined at the point where 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
Incorrect Use of Quotient Rule
While differentiating functions such as 4/(x²), students misapply the quotient rule. For example: Incorrect differentiation: d/dx (4/x²) = -8/x³. d/dx (u/v) = (v·u' - u·v') / v² (where u = 4 and v = x²)
Applying the quotient rule, d/dx (4/x²) = (x²·0 - 4·2x) / x⁴ To avoid this mistake, write the quotient rule without errors.
Always check for errors in the calculation and ensure it is properly simplified.
Mistake 4
Not Writing Constants and Coefficients
There is a common mistake that students at times forget to multiply the constants placed before x. For example, they incorrectly write d/dx (4/x) = 1/x².
Students should check the constants in the terms and ensure they are multiplied properly. For e.g., the correct equation is d/dx (4/x) = -4/x².
Mistake 5
Ignoring Negative Exponents
Students often forget to apply the power rule correctly with negative exponents.
For example: Incorrect: d/dx (x⁻¹) = 1/x. To fix this error, students should apply the power rule properly.
The correct differentiation is d/dx (x⁻¹) = -x⁻².
Examples Using the Derivative of 4/x
Problem 1
Calculate the derivative of (4/x)·x³.
Here, we have f(x) = (4/x)·x³. Using the product rule, f'(x) = u′v + uv′ In the given equation, u = 4/x and v = x³. Let’s differentiate each term, u′ = d/dx (4/x) = -4/x² v′ = d/dx (x³) = 3x²
Substituting into the given equation, f'(x) = (-4/x²)·x³ + (4/x)·3x²
Let’s simplify terms to get the final answer, f'(x) = -4x + 12
Thus, the derivative of the specified function is -4x + 12.
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.
Problem 2
A water tank is being emptied, and the rate at which the water level decreases is represented by y = 4/x, where y represents the rate of decrease in liters per minute, and x is time in minutes. If x = 2 minutes, find the rate of change of this rate.
We have y = 4/x (rate of water level decrease)...(1) Now, we will differentiate the equation (1)
Take the derivative of 4/x: dy/dx = -4/x² Given x = 2 (substitute this into the derivative) dy/dx = -4/(2)² = -1
Hence, the rate of change of the rate of decrease is -1 liter per minute per minute at x = 2 minutes.
Explanation
We find the rate of change of the rate of decrease at x = 2 minutes as -1, which means that the decrease in water level slows down at that point.
Problem 3
Derive the second derivative of the function y = 4/x.
The first step is to find the first derivative, dy/dx = -4/x²...(1) Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [-4/x²] Using the power rule, d²y/dx² = 8/x³ Therefore, the second derivative of the function y = 4/x is 8/x³.
Explanation
We use the step-by-step process, starting with the first derivative. Using the power rule, we differentiate -4/x². We then simplify the terms to find the final answer.
Problem 4
Prove: d/dx (2(4/x)) = -8/x².
Let’s start using the constant multiple rule: Consider y = 2(4/x) = 8/x
To differentiate, we use the power rule: dy/dx = -8/x² Hence, proved.
Explanation
In this step-by-step process, we used the constant multiple rule to differentiate the equation.
Then, we applied the power rule to find the derivative.
Problem 5
Solve: d/dx (4/(x + 1))
To differentiate the function, we use the quotient rule: d/dx (4/(x + 1)) = (d/dx (4)·(x + 1) - 4·d/dx (x + 1))/ (x + 1)²
We will substitute d/dx (4) = 0 and d/dx (x + 1) = 1 = (0·(x + 1) - 4·1)/ (x + 1)² = -4/(x + 1)²
Therefore, d/dx (4/(x + 1)) = -4/(x + 1)².
Explanation
In this process, we differentiate the given function using the quotient rule. As a final step, we simplify the equation to obtain the final result.
FAQs on the Derivative of 4/x
1.Find the derivative of 4/x.
2.Can we use the derivative of 4/x in real life?
3.Is it possible to take the derivative of 4/x at the point where x = 0?
4.What rule is used to differentiate 4/(x + 1)?
5.Are the derivatives of 4/x and 1/x the same?
Important Glossaries for the Derivative of 4/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 k/x, where k is a constant.
- Power Rule: A basic differentiation rule used to find the derivative of x raised to a power.
- Quotient Rule: A rule used to differentiate functions expressed as the quotient of two other functions.
- Undefined Points: Points at which a function does not exist or is not continuous.
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.
Fun Fact
: He loves to play the quiz with kids through algebra to make kids love it.
