100 LearnersLast updated on July 21st, 2025
Derivative of Arctan(x/2)
We use the derivative of arctan(x/2), which is 1/(1 + (x/2)²) * 1/2, as a measuring tool for how the arctangent 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 arctan(x/2) in detail.
What is the Derivative of Arctan(x/2)?
We now understand the derivative of arctan(x/2). It is commonly represented as d/dx (arctan(x/2)) or (arctan(x/2))', and its value is 1/(1 + (x/2)²) * 1/2. The function arctan(x/2) has a clearly defined derivative, indicating it is differentiable within its domain. The key concepts are mentioned below: Arctangent Function: arctan(x) is the inverse of the tangent function. Chain Rule: Rule for differentiating composite functions like arctan(x/2). Quotient Rule: Useful in deriving the derivative when applicable.
Derivative of Arctan(x/2) Formula
The derivative of arctan(x/2) can be denoted as d/dx (arctan(x/2)) or (arctan(x/2))'. The formula we use to differentiate arctan(x/2) is: d/dx (arctan(x/2)) = 1/(1 + (x/2)²) * 1/2 The formula applies to all x.
Proofs of the Derivative of Arctan(x/2)
We can derive the derivative of arctan(x/2) using proofs. To show this, we will use the trigonometric identities along with the rules of differentiation. There are several methods we use to prove this, such as: Using Chain Rule We will now demonstrate that the differentiation of arctan(x/2) results in 1/(1 + (x/2)²) * 1/2 using the chain rule: Using Chain Rule To prove the differentiation of arctan(x/2) using the chain rule, We use the formula: Arctan(x/2) = arctan(u) where u = x/2 Let y = arctan(u), then dy/du = 1/(1 + u²) Also, du/dx = 1/2 By the chain rule: dy/dx = dy/du * du/dx Substitute these values: dy/dx = 1/(1 + (x/2)²) * 1/2 Hence, proved.
Higher-Order Derivatives of Arctan(x/2)
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(x/2). 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(x/2), 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 of arctan(x/2) = 1/(1 + 0) * 1/2 = 1/2. The function is defined for all real x, so there are no points of discontinuity for the derivative.
Common Mistakes and How to Avoid Them in Derivatives of Arctan(x/2)
Students frequently make mistakes when differentiating arctan(x/2). 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 product or chain 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 to Apply the Chain Rule
They might not remember to apply the chain rule correctly when differentiating arctan(x/2). The inner function x/2 needs to be differentiated separately. Ensure you correctly apply the chain rule by finding the derivative of the inner function.
Mistake 3
Incorrect Use of the Chain Rule
While differentiating functions such as arctan(g(x)), students misapply the chain rule. For example: Incorrect differentiation: d/dx (arctan(x/2)) = 1/(1 + x²). The correct application is to differentiate the inner function as well: d/dx (arctan(x/2)) = 1/(1 + (x/2)²) * 1/2.
Mistake 4
Not Writing Constants and Coefficients
There is a common mistake that students at times forget to multiply the constants, such as the 1/2 in arctan(x/2). For example, they incorrectly write d/dx (arctan(x/2)) = 1/(1 + (x/2)²). Always check for constants and ensure they are multiplied properly. For example, the correct derivative is 1/(1 + (x/2)²) * 1/2.
Mistake 5
Confusing Arctan and Tan Derivatives
Students often confuse the derivatives of arctan(x) and tan(x). Remember, the derivative of arctan(x) is 1/(1 + x²), while the derivative of tan(x) is sec²x. Ensure you are using the correct derivative for arctan(x/2).
Examples Using the Derivative of Arctan(x/2)
Problem 1
Calculate the derivative of (arctan(x/2) · ln(x)).
Here, we have f(x) = arctan(x/2) · ln(x). Using the product rule, f'(x) = u′v + uv′ In the given equation, u = arctan(x/2) and v = ln(x). Let’s differentiate each term, u′= d/dx (arctan(x/2)) = 1/(1 + (x/2)²) * 1/2 v′= d/dx (ln(x)) = 1/x Substituting into the given equation, f'(x) = (1/(1 + (x/2)²) * 1/2) · ln(x) + (arctan(x/2)) · (1/x) Let’s simplify terms to get the final answer, f'(x) = (ln(x)/(2(1 + (x/2)²))) + (arctan(x/2)/x) Thus, the derivative of the specified function is (ln(x)/(2(1 + (x/2)²))) + (arctan(x/2)/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 new technology company is analyzing the growth of their user base over time. The growth is represented by the function y = arctan(x/2) where y represents user engagement, and x is time in months. If x = 2 months, calculate the rate of change of user engagement.
We have y = arctan(x/2) (user engagement growth)...(1) Now, we will differentiate the equation (1) Take the derivative: dy/dx = 1/(1 + (x/2)²) * 1/2 Given x = 2 (substitute this into the derivative) dy/dx = 1/(1 + (2/2)²) * 1/2 dy/dx = 1/(1 + 1) * 1/2 = 1/4 Hence, we get the rate of change of user engagement at x = 2 months as 1/4.
Explanation
We find the rate of change of user engagement at x = 2 months as 1/4, indicating that at this point, the engagement is increasing at a quarter of the rate per unit change in time.
Problem 3
Derive the second derivative of the function y = arctan(x/2).
The first step is to find the first derivative, dy/dx = 1/(1 + (x/2)²) * 1/2...(1) Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [1/(1 + (x/2)²) * 1/2] Here we use the quotient and chain rules, d²y/dx² = -1/2 * (d/dx [(1 + (x/2)²)⁻¹]) = -1/2 * [-(1 + (x/2)²)⁻² * (x/2)] * (1/2) = (x/2)/(1 + (x/2)²)² Therefore, the second derivative of the function y = arctan(x/2) is (x/2)/(1 + (x/2)²)².
Explanation
We use the step-by-step process, where we start with the first derivative. Using the quotient and chain rules, we differentiate the expression. We then simplify the terms to find the final answer.
Problem 4
Prove: d/dx (arctan²(x/2)) = arctan(x/2) * 1/(1 + (x/2)²).
Let’s start using the chain rule: Consider y = arctan²(x/2) = [arctan(x/2)]² To differentiate, we use the chain rule: dy/dx = 2 * arctan(x/2) * d/dx [arctan(x/2)] Since the derivative of arctan(x/2) is 1/(1 + (x/2)²) * 1/2, dy/dx = 2 * arctan(x/2) * 1/(1 + (x/2)²) * 1/2 = arctan(x/2) * 1/(1 + (x/2)²) Hence proved.
Explanation
In this step-by-step process, we used the chain rule to differentiate the equation. Then, we replace arctan(x/2) with its derivative. As a final step, we simplify the expression to derive the equation.
Problem 5
Solve: d/dx (arctan(x/2)/x).
To differentiate the function, we use the quotient rule: d/dx (arctan(x/2)/x) = (d/dx (arctan(x/2)) * x - arctan(x/2) * d/dx(x))/x² We will substitute d/dx (arctan(x/2)) = 1/(1 + (x/2)²) * 1/2 and d/dx(x) = 1 = (1/(1 + (x/2)²) * 1/2 * x - arctan(x/2))/x² = (x/(2(1 + (x/2)²)) - arctan(x/2))/x² Therefore, d/dx (arctan(x/2)/x) = (x/(2(1 + (x/2)²)) - arctan(x/2))/x².
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 Arctan(x/2)
1.Find the derivative of arctan(x/2).
2.Can we use the derivative of arctan(x/2) in real life?
3.Is it possible to take the derivative of arctan(x/2) at the point where x = π/2?
4.What rule is used to differentiate arctan(x/2)?
5.Are the derivatives of arctan(x/2) and arctan(x) the same?
Important Glossaries for the Derivative of Arctan(x/2)
Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x. Arctangent Function: The inverse of the tangent function, specifically focused on here as arctan(x/2). Chain Rule: A fundamental rule in calculus used to differentiate composite functions. Quotient Rule: A rule used to differentiate functions that are the ratio of two differentiable functions. First Derivative: The initial result of differentiation, indicating the rate of change of a function. ```
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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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