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106 LearnersLast updated on September 12, 2025
Derivative of 1/r
We use the derivative of 1/r, which is -1/r², as a measuring tool to understand how the function changes in response to a slight change in r. Derivatives help us calculate profit or loss in real-life situations. We will now talk about the derivative of 1/r in detail.
What is the Derivative of 1/r?
We now understand the derivative of 1/r. It is commonly represented as d/dr (1/r) or (1/r)', and its value is -1/r². The function 1/r has a clearly defined derivative, indicating it is differentiable within its domain.
The key concepts are mentioned below:
Reciprocal Function: (1/r) is the reciprocal of r.
Power Rule: Rule for differentiating r raised to a power.
Negative Exponent: 1/r can be expressed as r^(-1).
Derivative of 1/r Formula
The derivative of 1/r can be denoted as d/dr (1/r) or (1/r)'.
The formula we use to differentiate 1/r is: d/dr (1/r) = -1/r² (or) (1/r)' = -1/r²
The formula applies to all r where r ≠ 0.
Proofs of the Derivative of 1/r
We can derive the derivative of 1/r using proofs. To show this, we will use the 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
We will now demonstrate that the differentiation of 1/r results in -1/r² using the above-mentioned methods:
By First Principle
The derivative of 1/r can be proved using the First Principle, which expresses the derivative as the limit of the difference quotient.
To find the derivative of 1/r using the first principle, we will consider f(r) = 1/r.
Its derivative can be expressed as the following limit. f'(r) = limₕ→₀ [f(r + h) - f(r)] / h … (1)
Given that f(r) = 1/r, we write f(r + h) = 1/(r + h).
Substituting these into equation (1),
f'(r) = limₕ→₀ [1/(r + h) - 1/r] / h = limₕ→₀ [(r - (r + h)) / (r(r + h))] / h = limₕ→₀ [-h / (r² + rh)] / h = limₕ→₀ [-1 / (r² + rh)]
As h approaches 0, the term rh vanishes, resulting in: f'(r) = -1/r²
Hence, proved.
Using Power Rule
To prove the differentiation of 1/r using the power rule, We express 1/r as r(-1). Consider f(r) = r(-1)
Using the power rule for differentiation: f'(r) = -1 * r(-2) = -1/r²
Thus, by applying the power rule, we find that the derivative is -1/r².
Higher-Order Derivatives of 1/r
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 1/r.
For the first derivative of a function, we write f′(r), 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′′(r). Similarly, the third derivative, f′′′(r), is the result of the second derivative and this pattern continues.
For the nth Derivative of 1/r, we generally use fⁿ(r) for the nth derivative of a function f(r), which tells us the change in the rate of change (continuing for higher-order derivatives).
Special Cases:
When r is 0, the derivative is undefined because 1/r has a vertical asymptote there. When r is 1, the derivative of 1/r = -1/1², which is -1.
Common Mistakes and How to Avoid Them in Derivatives of 1/r
Students frequently make mistakes when differentiating 1/r. 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 expression
Students may forget to simplify the expression, which can lead to incomplete or incorrect results. They often skip steps and directly arrive at the result, especially when solving using the power 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 1/r
They might not remember that 1/r is undefined at the point r = 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 Power Rule
While differentiating functions such as 1/r², students misapply the power rule. For example: Incorrect differentiation: d/dr (1/r²) = -2/r³. Using the power rule: d/dr (r^(-2)) = (-2)r^(-3) = -2/r³. To avoid this mistake, follow the power rule carefully and ensure proper application.
Mistake 4
Not writing Constants and Coefficients
There is a common mistake that students at times forget to multiply the constants placed before 1/r. For example, they incorrectly write d/dr (5/r) = -1/r². Students should check the constants in the terms and ensure they are multiplied properly. For e.g., the correct equation is d/dr (5/r) = -5/r².
Mistake 5
Not Applying the Chain Rule
Students often forget to use the chain rule. This happens when the derivative of the inner function is not considered. For example: Incorrect: d/dr (1/(2r)) = -1/(2r)². To fix this error, students should divide the functions into inner and outer parts. Then, make sure that each function is differentiated. For example, d/dr (1/(2r)) = -1/(2r)² * 2.
Examples Using the Derivative of 1/r
Problem 1
Calculate the derivative of (1/r · r²).
Here, we have f(r) = (1/r) · r². Using simplification, f(r) = r. The derivative of r is, f'(r) = 1. Thus, the derivative of the specified function is 1.
Explanation
We find the derivative of the given function by simplifying the function first and then differentiating the simplified function to get the final result.
Problem 2
A spherical balloon is being inflated, and its radius r increases as a function of time t. The volume V is inversely proportional to r, given by V = k/r. If the radius r is 5 meters, find the rate of change of the volume with respect to r.
We have V = k/r (volume of the balloon)...(1)
Now, we will differentiate the equation (1).
Take the derivative with respect to r: dV/dr = -k/r².
Given r = 5 meters (substitute this into the derivative),
dV/dr = -k/5². dV/dr = -k/25.
Hence, the rate of change of the volume with respect to r when r = 5 meters is -k/25.
Explanation
We find the rate of change of the volume at r = 5 meters, which indicates that as the radius increases, the volume decreases at a rate proportional to 1/r².
Problem 3
Derive the second derivative of the function V = 1/r.
The first step is to find the first derivative, dV/dr = -1/r²...(1)
Now we will differentiate equation (1) to get the second derivative: d²V/dr² = d/dr [-1/r²]
Using the power rule, d²V/dr² = 2/r³.
Therefore, the second derivative of the function V = 1/r is 2/r³.
Explanation
We use the step-by-step process, where we start with the first derivative. Using the power rule, we differentiate -1/r² to obtain the second derivative.
Problem 4
Prove: d/dr (1/r²) = -2/r³.
Let’s start using the power rule: Consider V = 1/r² = r(-2)
To differentiate, we use the power rule: dV/dr = -2r(-3) = -2/r³
Hence, proved.
Explanation
In this step-by-step process, we used the power rule to differentiate the equation, which involves multiplying by the exponent and reducing the power by 1.
Problem 5
Solve: d/dr (1/r + r).
To differentiate the function, we use basic differentiation rules:
d/dr (1/r + r) = d/dr (1/r) + d/dr (r) = -1/r² + 1.
Therefore, d/dr (1/r + r) = -1/r² + 1.
Explanation
In this process, we differentiate each term separately and then combine them to obtain the final result.
FAQs on the Derivative of 1/r
1.Find the derivative of 1/r.
2.Can we use the derivative of 1/r in real life?
3.Is it possible to take the derivative of 1/r at the point where r = 0?
4.What rule is used to differentiate 1/r²?
5.Are the derivatives of 1/r and r^(-1) the same?
Important Glossaries for the Derivative of 1/r
- Derivative: The derivative of a function indicates how the given function changes in response to a slight change in r.
- Reciprocal Function: A function of the form 1/r, where r is a variable.
- Power Rule: A basic rule in calculus used to find the derivative of a function of the form rn.
- Undefined Point: A point where a function does not have a value, such as r = 0 for 1/r.
- Rate of Change: A measure of how a quantity changes with respect to another quantity, often time or space.
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.
