102 LearnersLast updated on July 18th, 2025
Derivative of Arctan 4x
We use the derivative of arctan(4x), which helps us understand how the arctangent function changes in response to a slight change in x. Derivatives are useful in various real-life applications, such as calculating rates of change. We will now discuss the derivative of arctan(4x) in detail.
What is the Derivative of Arctan 4x?
We now explore the derivative of arctan(4x). It is commonly represented as d/dx (arctan(4x)) or (arctan(4x))', and its value is 4/(1+(4x)²). The function arctan(4x) has a well-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: A rule for differentiating composite functions. Inverse Trigonometric Functions: Functions that reverse the effect of the trigonometric functions.
Derivative of Arctan 4x Formula
The derivative of arctan(4x) can be denoted as d/dx (arctan(4x)) or (arctan(4x))'. The formula we use to differentiate arctan(4x) is: d/dx (arctan(4x)) = 4/(1+(4x)²) The formula applies to all x where the expression 1+(4x)² is nonzero.
Proofs of the Derivative of Arctan 4x
We can derive the derivative of arctan(4x) 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 Chain Rule To prove the differentiation of arctan(4x) using the chain rule, We use the formula for the derivative of arctan(x): d/dx (arctan(x)) = 1/(1+x²) Consider u = 4x; hence, arctan(4x) = arctan(u). By the chain rule: d/dx (arctan(u)) = (d/du (arctan(u))) * (du/dx) d/dx (arctan(4x)) = (1/(1+u²)) * 4 Substituting u = 4x, we get: d/dx (arctan(4x)) = 4/(1+(4x)²)
Higher-Order Derivatives of Arctan 4x
When a function is differentiated multiple times, the derivatives obtained are referred to as higher-order derivatives. Higher-order derivatives can be a bit complex. 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(4x). 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(4x), we generally use fⁿ(x) for the nth derivative of a function f(x), which tells us the change in the rate of change.
Special Cases:
When x is ±∞, the derivative approaches 0 because the function levels out as it approaches its horizontal asymptotes. When x is 0, the derivative of arctan(4x) = 4/(1+0²) = 4.
Common Mistakes and How to Avoid Them in Derivatives of Arctan 4x
Students frequently make mistakes when differentiating arctan(4x). 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 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 the Domain of Arctan 4x
They might not remember that the domain of arctan(x) is all real numbers, but the expression 1+(4x)² must be nonzero. 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 Chain Rule
While differentiating functions such as arctan(4x), students misapply the chain rule. For example: Incorrect differentiation: d/dx (arctan(4x)) = 1/(1+(4x)²). To correct this, remember to apply the chain rule properly: d/dx (arctan(4x)) = 4/(1+(4x)²). To avoid this mistake, 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 sometimes forget to multiply the constants placed before x. For example, they incorrectly write d/dx (arctan(4x)) = 1/(1+(4x)²). Students should check the constants in the terms and ensure they are multiplied properly. For example, the correct equation is d/dx (arctan(4x)) = 4/(1+(4x)²).
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/dx (arctan(2x)) = 1/(1+(2x)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/dx (arctan(2x)) = 2/(1+(2x)²).
Examples Using the Derivative of Arctan 4x
Problem 1
Calculate the derivative of arctan(2x)·4x²
Here, we have f(x) = arctan(2x)·4x². Using the product rule, f'(x) = u′v + uv′ In the given equation, u = arctan(2x) and v = 4x². Let’s differentiate each term, u′ = d/dx (arctan(2x)) = 2/(1+(2x)²) v′ = d/dx (4x²) = 8x Substituting into the given equation, f'(x) = (2/(1+(2x)²))·(4x²) + (arctan(2x))·8x Let’s simplify terms to get the final answer, f'(x) = (8x²/(1+(2x)²)) + 8x·arctan(2x) Thus, the derivative of the specified function is (8x²/(1+(2x)²)) + 8x·arctan(2x).
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 is modeling the angle of rotation θ(t) of a robotic arm using the function θ(t) = arctan(4t), where t is time in seconds. Find the rate of change of the angle at t = 1 second.
We have θ(t) = arctan(4t) (angle of the robotic arm)...(1) Now, we will differentiate the equation (1) Take the derivative arctan(4t): dθ/dt = 4/(1+(4t)²) Given t = 1 (substitute this into the derivative) dθ/dt = 4/(1+(4(1))²) = 4/(1+16) = 4/17 Hence, we get the rate of change of the angle at t = 1 second as 4/17 radians per second.
Explanation
We find the rate of change of the angle at t = 1 second as 4/17, which means that at a given point, the angle of the robotic arm changes at this rate with respect to time.
Problem 3
Derive the second derivative of the function y = arctan(4x).
The first step is to find the first derivative, dy/dx = 4/(1+(4x)²)...(1) Now we will differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [4/(1+(4x)²)] Using the chain rule and quotient rule, d²y/dx² = -32x/(1+(4x)²)² Therefore, the second derivative of the function y = arctan(4x) is -32x/(1+(4x)²)².
Explanation
We use the step-by-step process, where we start with the first derivative. Using the chain and quotient rule, we differentiate 4/(1+(4x)²). We then substitute the identity and simplify the terms to find the final answer.
Problem 4
Prove: d/dx (arctan²(x)) = 2arctan(x)/(1+x²).
Let’s start using the chain rule: Consider y = arctan²(x) [arctan(x)]² To differentiate, we use the chain rule: dy/dx = 2arctan(x)·d/dx [arctan(x)] Since the derivative of arctan(x) is 1/(1+x²), dy/dx = 2arctan(x)·(1/(1+x²)) Substituting y = arctan²(x), d/dx (arctan²(x)) = 2arctan(x)/(1+x²) Hence proved.
Explanation
In this step-by-step process, we used the chain rule to differentiate the equation. Then, we replace arctan(x) with its derivative. As a final step, we substitute y = arctan²(x) to derive the equation.
Problem 5
Solve: d/dx (arctan(3x)/x)
To differentiate the function, we use the quotient rule: d/dx (arctan(3x)/x) = [(d/dx (arctan(3x)).x - arctan(3x).d/dx(x))]/x² We will substitute d/dx (arctan(3x)) = 3/(1+(3x)²) and d/dx (x) = 1 = [(3/(1+(3x)²)).x - arctan(3x)]/x² = [3x/(1+(3x)²) - arctan(3x)]/x² Therefore, d/dx (arctan(3x)/x) = [3x/(1+(3x)²) - arctan(3x)]/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 4x
1.Find the derivative of arctan(4x).
2.Can we use the derivative of arctan(4x) in real life?
3.Is it possible to take the derivative of arctan(4x) when x = ±∞?
4.What rule is used to differentiate arctan(4x)/x?
5.Are the derivatives of arctan(x) and arctan(4x) the same?
Important Glossaries for the Derivative of Arctan 4x
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 the inverse of the tangent function and is denoted as arctan(x). Chain Rule: A differentiation rule used to differentiate composite functions. Inverse Trigonometric Functions: Functions that reverse the effect of trigonometric functions, such as arctan(x). Quotient Rule: A rule used for differentiating functions that are ratios of two differentiable functions.
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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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