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108 LearnersLast updated on October 8, 2025
Derivative of Arctan(x²)
We use the derivative of arctan(x), which is 1/(1 + x²), 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 discuss the derivative of arctan(x²) in detail.
What is the Derivative of Arctan(x²)?
We now understand the derivative of arctan(x²). It is commonly represented as d/dx (arctan(x²)) or (arctan(x²))', and its value is 2x/(1 + x⁴). The function arctan(x²) has a clearly defined derivative, indicating it is differentiable within its domain.
The key concepts are mentioned below:
Arctangent Function: arctan(u) where u = x².
Chain Rule: Used for differentiating composite functions like arctan(x²).
Derivative of Arctan: d/dx (arctan(x)) = 1/(1 + x²).
Derivative of Arctan(x²) Formula
The derivative of arctan(x²) can be denoted as d/dx (arctan(x²)) or (arctan(x²))'.
The formula we use to differentiate arctan(x²) is: d/dx (arctan(x²)) = 2x/(1 + x⁴).
The formula applies to all x where the expression is defined.
Proofs of the Derivative of Arctan(x²)
We can derive the derivative of arctan(x²) using proofs. To show this, we will use the chain rule along with the rules of differentiation.
There are several methods we use to prove this, such as:
- By Chain Rule
- By First Principles
Using Chain Rule
To prove the differentiation of arctan(x²) using the chain rule: Let u = x², then arctan(x²) = arctan(u). We use the formula: d/dx (arctan(u)) = 1/(1 + u²) · du/dx. So, d/dx (arctan(x²)) = 1/(1 + (x²)²) · 2x = 2x/(1 + x⁴). Hence, proved.
Using First Principles
The derivative of arctan(x²) can also be proved using the First Principle, which expresses the derivative as the limit of the difference quotient: f(x) = arctan(x²), so f'(x) = limₕ→₀ [arctan((x + h)²) - arctan(x²)] / h. The proof follows by applying trigonometric identities and calculating limits, ultimately leading to the derivative 2x/(1 + x⁴).
Higher-Order Derivatives of Arctan(x²)
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(x²).
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(x²), 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(x²) = 2 · 0/(1 + 0) = 0.
When x is any value where x⁴ = -1, the derivative is undefined because the denominator becomes zero, but such x does not exist for real numbers.
Common Mistakes and How to Avoid Them in Derivatives of Arctan(x²)
Students frequently make mistakes when differentiating arctan(x²). 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. They often skip steps and directly arrive at the result.
Ensure that the derivative of the inner function (x²) is calculated and multiplied properly.
Mistake 2
Forgetting to square the x² in the denominator
They might not correctly square the inner function in the denominator, leading to incorrect results.
Always remember that the arctan(x²) differentiation involves (x²)² in the denominator.
Mistake 3
Incorrect use of trigonometric identities
While differentiating, students may misapply trigonometric identities. For example, incorrectly writing the derivative of arctan(x) instead of arctan(x²).
Always ensure the correct identities are applied for arctan(x²).
Mistake 4
Ignoring the need for simplification
Students often forget to simplify the derivative, resulting in a more complicated form than necessary.
Ensure that you simplify the expression to reach the standard form of the derivative.
Mistake 5
Not identifying the domain restrictions
Students might not recognize the domain restrictions where the derivative is undefined.
It's important to verify where the denominator might be zero to understand where the derivative is valid.
Examples Using the Derivative of Arctan(x²)
Problem 1
Calculate the derivative of (arctan(x²) · e^x).
Here, we have f(x) = arctan(x²) · e^x. Using the product rule, f'(x) = u′v + uv′. In the given equation, u = arctan(x²) and v = e^x. Let’s differentiate each term, u′ = d/dx (arctan(x²)) = 2x/(1 + x⁴), v′ = d/dx (e^x) = e^x. Substituting into the given equation, f'(x) = (2x/(1 + x⁴)) · e^x + arctan(x²) · e^x. Let’s simplify terms to get the final answer, f'(x) = (2x e^x)/(1 + x⁴) + arctan(x²) · e^x. Thus, the derivative of the specified function is (2x e^x)/(1 + x⁴) + arctan(x²) · e^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 roller coaster is designed with a track elevation represented by the function y = arctan(x²), where y represents the height of the track at a distance x. If x = 1 meter, measure the slope of the track.
We have y = arctan(x²) (slope of the track)...(1). Now, we will differentiate equation (1). Take the derivative arctan(x²): dy/dx = 2x/(1 + x⁴). Given x = 1, substitute this into the derivative: dy/dx = 2(1)/(1 + 1⁴) = 2/2 = 1. Hence, we get the slope of the track at a distance x = 1 as 1.
Explanation
We find the slope of the track at x = 1 as 1, which means that at a given point, the height of the track would rise at a rate equal to the horizontal distance.
Problem 3
Derive the second derivative of the function y = arctan(x²).
The first step is to find the first derivative, dy/dx = 2x/(1 + x⁴)...(1). Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [2x/(1 + x⁴)]. Here we use the quotient rule, d²y/dx² = [(1 + x⁴)·2 - 2x·4x³]/(1 + x⁴)², = [2 + 2x⁴ - 8x⁴]/(1 + x⁴)², = [2 - 6x⁴]/(1 + x⁴)². Therefore, the second derivative of the function y = arctan(x²) is [2 - 6x⁴]/(1 + x⁴)².
Explanation
We use the step-by-step process, where we start with the first derivative.
Using the quotient rule, we differentiate 2x/(1 + x⁴).
We then simplify the terms to find the final answer.
Problem 4
Prove: d/dx (x · arctan(x²)) = arctan(x²) + 2x²/(1 + x⁴).
Let’s start using the product rule: Consider y = x · arctan(x²). To differentiate, we use the product rule: dy/dx = x' · arctan(x²) + x · (d/dx [arctan(x²)]). Since the derivative of arctan(x²) is 2x/(1 + x⁴), dy/dx = 1 · arctan(x²) + x · (2x/(1 + x⁴)), = arctan(x²) + 2x²/(1 + x⁴). Hence proved.
Explanation
In this step-by-step process, we used the product rule to differentiate the equation.
Then, we replace arctan(x²) with its derivative.
As a final step, we substitute y = x · arctan(x²) to derive the equation.
Problem 5
Solve: d/dx (arctan(x²)/x).
To differentiate the function, we use the quotient rule: d/dx (arctan(x²)/x) = (d/dx (arctan(x²)) · x - arctan(x²) · d/dx(x))/x². We will substitute d/dx (arctan(x²)) = 2x/(1 + x⁴) and d/dx (x) = 1, = [(2x/(1 + x⁴)) · x - arctan(x²)]/x², = [2x²/(1 + x⁴) - arctan(x²)]/x². Therefore, d/dx (arctan(x²)/x) = [2x²/(1 + x⁴) - arctan(x²)]/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(x²)
1.Find the derivative of arctan(x²).
2.Can we use the derivative of arctan(x²) in real life?
3.Is it possible to take the derivative of arctan(x²) at the point where x = 0?
4.What rule is used to differentiate arctan(x²)/x?
5.Are the derivatives of arctan(x²) and arctan(x) the same?
Important Glossaries for the Derivative of Arctan(x²)
- Derivative: The derivative of a function indicates how the given function changes in response to a slight change in x.
- Arctangent Function: The arctangent function is one of the inverse trigonometric functions and is written as arctan(x).
- Chain Rule: A rule for differentiating composite functions like arctan(x²).
- Quotient Rule: A method for differentiating functions that are divided by one another.
- Higher-Order Derivatives: These derivatives indicate the rate of change of the rate of change of a function.
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Jaskaran Singh Saluja
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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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