101 LearnersLast updated on July 24th, 2025
Derivative of -4/x
We use the derivative of -4/x, which is 4/x², as a tool to understand 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 -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 can be considered as a reciprocal function. Power Rule: A rule for differentiating functions of the form xⁿ. Negative Coefficient: The function -4/x has a negative coefficient that impacts its derivative.
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 (or) (-4/x)' = 4/x² The formula applies to all x where 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. Here are some methods we use to prove this: By Power Rule The function -4/x can be expressed as -4x⁻¹. Differentiate using the power rule: d/dx (xⁿ) = n*xⁿ⁻¹. For -4x⁻¹, n = -1: d/dx (-4x⁻¹) = (-1)(-4)x⁻² = 4x⁻² = 4/x². Hence, proved. By Quotient Rule Consider u = -4 and v = x. Apply the quotient rule: d/dx (u/v) = (v * du/dx - u * dv/dx) / v². Here, du/dx = 0 and dv/dx = 1: d/dx (-4/x) = (x*0 - (-4)*1) / x² = 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 how the acceleration of a car (second derivative) changes as the speed (first derivative) 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 = 0, the derivative is undefined because -4/x has a vertical asymptote there. When x is any non-zero real number, the derivative of -4/x = 4/x².
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 basic differentiation rules. 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 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 certain points.
Mistake 3
Incorrect use of Power Rule
While differentiating functions such as -4/x, students misapply the power rule. For example: Incorrect differentiation: d/dx (-4x⁻¹) = -4x⁻². To avoid this mistake, remember that the power rule requires multiplying by the power and reducing the power by one. 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) = x⁻². Students should check the constants in the terms and ensure they are multiplied properly. For example, the correct equation is d/dx (-4/x) = 4/x².
Mistake 5
Not Applying the Quotient Rule Correctly
Students often forget to use the quotient rule correctly. This happens when both the numerator and denominator are not properly differentiated. For example: Incorrect: d/dx (-4/x) = 0 - 4/x². To fix this error, students should use the quotient rule: (v*u' - u*v')/v². For example, d/dx (-4/x) = 4/x².
Examples Using the Derivative of -4/x
Problem 1
Calculate the derivative of (-4/x)².
Here, we have f(x) = (-4/x)². Using the chain rule, f'(x) = 2(-4/x)(d/dx(-4/x)). In the given equation, d/dx(-4/x) = 4/x². Let’s substitute into the equation, f'(x) = 2(-4/x)(4/x²) = -32/x³. Thus, the derivative of the specified function is -32/x³.
Explanation
We find the derivative of the given function by first using the chain rule. The first step is finding the derivative of the inner function and then multiplying by the derivative of the outer function.
Problem 2
XYZ Corporation is analyzing the rate of change of their investment value represented by the function y = -4/x, where y represents the value at time x. If x = 2 years, find the rate of change of the investment.
We have y = -4/x (investment value)...(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 = 4/4 = 1. Hence, the rate of change of the investment at x = 2 years is 1.
Explanation
We find the rate of change of the investment at x = 2 years as 1, which means that at this time, the value of the investment changes at a constant rate.
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]. Consider: 4x⁻², d²y/dx² = -8x⁻³ = -8/x³. Therefore, the second derivative of the function y = -4/x is -8/x³.
Explanation
We use the step-by-step process, where we start 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 ((-4/x)²) = 32/x³.
Let’s start using the chain rule: Consider y = (-4/x)² = [-4/x]². To differentiate, we use the chain rule: dy/dx = 2(-4/x)(d/dx [-4/x]). Since the derivative of -4/x is 4/x², dy/dx = 2(-4/x)(4/x²) = -32/x³. Hence proved.
Explanation
In this step-by-step process, we used the chain rule to differentiate the equation. Then, we replace -4/x with its derivative. As a final step, we substitute y = (-4/x)² to derive the equation.
Problem 5
Solve: d/dx (-4x²/x).
To differentiate the function, we simplify the expression first: d/dx (-4x²/x) = d/dx (-4x). Now apply the power rule: d/dx (-4x) = -4. Therefore, d/dx (-4x²/x) = -4.
Explanation
In this process, we first simplify the given function to -4x and then differentiate using the power rule. The final step gives us the 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?
5.Can we find the second derivative of -4/x?
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 that is the inverse of another function, such as -4/x. Power Rule: A basic differentiation rule for functions of the form xⁿ. Undefined Point: A point where a function is not defined, such as x = 0 for -4/x. Chain Rule: A rule used to differentiate composite functions.
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.
