102 LearnersLast updated on July 23rd, 2025
Derivative of arctan(1/x)
We use the derivative of arctan(1/x), which is -1/(x² + 1), as a tool for understanding how the arctan(1/x) function changes with respect to a slight change in x. Derivatives help us calculate profit or loss in real-life situations. We will now discuss the derivative of arctan(1/x) in detail.
What is the Derivative of arctan(1/x)?
We now understand the derivative of arctan(1/x). It is commonly represented as d/dx (arctan(1/x)) or (arctan(1/x))', and its value is -1/(x² + 1).
The function arctan(1/x) has a clearly defined derivative, indicating it is differentiable within its domain.
The key concepts are mentioned below: Inverse Tangent Function: arctan(y) = x implies tan(x) = y.
Chain Rule: Rule for differentiating composite functions like arctan(1/x). Derivative of arctan(x): d/dx (arctan(x)) = 1/(1 + x²).
Derivative of arctan(1/x) Formula
The derivative of arctan(1/x) can be denoted as d/dx (arctan(1/x)) or (arctan(1/x))'.
The formula we use to differentiate arctan(1/x) is: d/dx (arctan(1/x)) = -1/(x² + 1)
The formula applies to all x where x ≠ 0.
Proofs of the Derivative of arctan(1/x)
We can derive the derivative of arctan(1/x) using proofs. To show this, we will use trigonometric identities along with the rules of differentiation.
There are several methods we use to prove this, such as: By First Principle Using Chain Rule Using Implicit Differentiation We will now demonstrate that the differentiation of arctan(1/x) results in -1/(x² + 1)
using the above-mentioned methods: Using Chain Rule To prove the differentiation of arctan(1/x) using the chain rule, We use the formula: u = 1/x and f(u) = arctan(u)
Thus, y = arctan(1/x) = arctan(u) By chain rule: dy/dx = dy/du * du/dx We know dy/du = 1/(1 + u²) and du/dx = -1/x² Substitute u = 1/x to get: dy/dx = 1/(1 + (1/x)²) * (-1/x²) = -1/(x² + 1)
Thus, the derivative is -1/(x² + 1). Using Implicit Differentiation Consider y = arctan(1/x).
Then, tan(y) = 1/x. Differentiating both sides with respect to x, using implicit differentiation: sec²(y) * dy/dx = -1/x² Since sec²(y) = 1 + tan²(y), substitute tan(y) = 1/x: (1 + (1/x)²) * dy/dx = -1/x² dy/dx = -1/(x² + 1)
Hence, proved.
Higher-Order Derivatives of arctan(1/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 arctan(1/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 arctan(1/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 arctan(1/x) has a vertical asymptote there. When x is 1, the derivative of arctan(1/x) = -1/(1² + 1) = -1/2.
Common Mistakes and How to Avoid Them in Derivatives of arctan(1/x)
Students frequently make mistakes when differentiating arctan(1/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 applying the chain rule correctly
Students may forget to apply the chain rule properly, which can lead to incorrect results.
Ensure that each step is written in order, especially when differentiating composite functions like arctan(1/x).
Mistake 2
Forgetting the undefined points of arctan(1/x)
They might not remember that arctan(1/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 such points.
Mistake 3
Incorrect use of trigonometric identities
While differentiating functions such as arctan(1/x), students misapply trigonometric identities.
For example, they might incorrectly assume the identity for arctan(x) applies directly without considering the chain rule. Clarify the steps and ensure correct application of identities.
Mistake 4
Not writing constants and coefficients
There is a common mistake where students forget to multiply constants or coefficients when differentiating.
For example, they might incorrectly write d/dx (5 * arctan(1/x)) as -1/(x² + 1) without the constant.
Always ensure constants are multiplied properly.
Mistake 5
Incorrect simplification of expressions
Simplifying expressions incorrectly after differentiation can lead to errors. Students might not combine terms properly or might overlook common factors. Review each step carefully to ensure correct simplification.
Examples Using the Derivative of arctan(1/x)
Problem 1
Calculate the derivative of arctan(1/x)·sin(x).
Here, we have f(x) = arctan(1/x)·sin(x).
Using the product rule, f'(x) = u′v + uv′ In the given equation, u = arctan(1/x) and v = sin(x). Let’s differentiate each term, u′ = d/dx (arctan(1/x)) = -1/(x² + 1) v′ = d/dx (sin(x)) = cos(x)
Substituting into the given equation, f'(x) = (-1/(x² + 1))·sin(x) + arctan(1/x)·cos(x)
Thus, the derivative of the specified function is (-sin(x)/(x² + 1)) + arctan(1/x)·cos(x).
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 company uses a sensor that measures the angle as arctan(1/x) for x meters away from the source. If x = 2 meters, find the rate of change of the angle.
We have y = arctan(1/x) (rate of change of the angle)...(1)
Now, we will differentiate the equation (1) Take the derivative of arctan(1/x): dy/dx = -1/(x² + 1)
Given x = 2, substitute this into the derivative: dy/dx = -1/(2² + 1) = -1/5
Hence, the rate of change of the angle at a distance x = 2 is -1/5.
Explanation
We find the rate of change of the angle at x = 2 as -1/5, meaning that for a small change in x near 2, the angle decreases at this rate.
Problem 3
Derive the second derivative of the function y = arctan(1/x).
The first step is to find the first derivative, dy/dx = -1/(x² + 1)...(1)
Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [-1/(x² + 1)]
Using the quotient rule, d²y/dx² = [0 * (x² + 1) + 1 * 2x] / (x² + 1)² d²y/dx² = 2x/(x² + 1)²
Therefore, the second derivative of the function y = arctan(1/x) is 2x/(x² + 1)².
Explanation
We use the step-by-step process, where we start with the first derivative.
Using the quotient rule, we differentiate -1/(x² + 1). We then simplify the terms to find the final answer.
Problem 4
Prove: d/dx (arctan(1/x)²) = -2 * arctan(1/x)/(x² + 1).
Let’s start using the chain rule: Consider y = (arctan(1/x))²
To differentiate, we use the chain rule: dy/dx = 2 * arctan(1/x) * d/dx [arctan(1/x)]
Since the derivative of arctan(1/x) is -1/(x² + 1), dy/dx = 2 * arctan(1/x) * (-1/(x² + 1))
Substituting y = (arctan(1/x))², d/dx (arctan(1/x)²) = -2 * arctan(1/x)/(x² + 1) Hence proved.
Explanation
In this step-by-step process, we used the chain rule to differentiate the equation.
Then, we replace arctan(1/x) with its derivative.
As a final step, we substitute y = (arctan(1/x))² to derive the equation.
Problem 5
Solve: d/dx (arctan(1/x)/x).
To differentiate the function, we use the quotient rule: d/dx (arctan(1/x)/x) = (d/dx (arctan(1/x)) * x - arctan(1/x) * d/dx(x))/x²
We will substitute d/dx (arctan(1/x)) = -1/(x² + 1) and d/dx(x) = 1 = (-1/(x² + 1) * x - arctan(1/x) * 1) / x² = (-x/(x² + 1) - arctan(1/x)) / x²
Therefore, d/dx (arctan(1/x)/x) = (-x/(x² + 1) - arctan(1/x)) / x²
Explanation
In this process, we differentiate the given function using the product rule and quotient rule. As a final step, we simplify the equation to obtain the final result.
FAQs on the Derivative of arctan(1/x)
1.Find the derivative of arctan(1/x).
2.Can we use the derivative of arctan(1/x) in real life?
3.Is it possible to take the derivative of arctan(1/x) at the point where x = 0?
4.What rule is used to differentiate arctan(1/x)/x?
5.Are the derivatives of arctan(1/x) and arctan(x) the same?
Important Glossaries for the Derivative of arctan(1/x)
- Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.
- Inverse Tangent Function: The inverse of the tangent function, represented as arctan(x), gives the angle whose tangent is x.
- Chain Rule: A fundamental rule for differentiating composite functions, allowing us to find the derivative of functions like arctan(1/x).
- Quotient Rule: A method used to differentiate functions that are divided by one another, such as arctan(1/x)/x.
- Asymptote: A line that a function approaches but never crosses, relevant for points where functions like arctan(1/x) become undefined.
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.
