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103 LearnersLast updated on September 15, 2025
Derivative of arctan(5x)
We use the derivative of arctan(5x), which is 5/(1 + (5x)²), to understand how the arctangent function changes with respect to x. Derivatives are essential in calculating rates of change in various applications. We will explore the derivative of arctan(5x) in detail.
What is the Derivative of arctan(5x)?
To understand the derivative of arctan(5x), we represent it as d/dx (arctan(5x)) or (arctan(5x))'. The value of the derivative is 5/(1 + (5x)²). The function arctan(5x) is differentiable across its domain. The key concepts include:
Arctangent Function: arctan(x) is the inverse of the tangent function.
Chain Rule: Used to differentiate composite functions.
Standard Derivative: d/dx (arctan(u)) = u'/(1 + u²).
Derivative of arctan(5x) Formula
The derivative of arctan(5x) is denoted as d/dx (arctan(5x)) or (arctan(5x))'.
The formula for differentiating arctan(5x) is: d/dx (arctan(5x)) = 5/(1 + (5x)²)
This formula applies to all x.
Proofs of the Derivative of arctan(5x)
We can prove the derivative of arctan(5x) using the chain rule. The proof involves differentiating the composite function using standard differentiation techniques. Here's how it's done:
Using Chain Rule
To prove the differentiation of arctan(5x) using the chain rule, we start with: Let u = 5x, then arctan(5x) = arctan(u).
The derivative of arctan(u) is d/dx (arctan(u)) = u'/(1 + u²).
Differentiating u = 5x gives u' = 5.
Substitute into the derivative formula: d/dx (arctan(5x)) = 5/(1 + (5x)²).
Hence, proved.
Higher-Order Derivatives of arctan(5x)
Higher-order derivatives of functions like arctan(5x) can provide insights into the function's behavior. Differentiating arctan(5x) multiple times yields higher-order derivatives.
For instance, the first derivative gives the rate of change, while the second derivative indicates how this rate changes.
For the first derivative, we write y′(x). The second derivative, y′′(x), is derived from the first derivative, and this pattern continues.
For the nth derivative, we use yⁿ(x).
Special Cases:
When x is 0, the derivative of arctan(5x) is 5/(1 + (5×0)²) = 5.
As x approaches infinity, the derivative approaches 0 because the denominator grows much larger than the numerator.
Common Mistakes and How to Avoid Them in Derivatives of arctan(5x)
Students often make mistakes when differentiating arctan(5x). Understanding the correct process can help avoid these errors. Here are some common mistakes and solutions:
Mistake 1
Incorrect Application of the Chain Rule
Students may forget to apply the chain rule correctly. It's essential to differentiate the inner function and multiply it by the derivative of the outer function. For example, incorrect: d/dx (arctan(5x)) = 1/(1 + (5x)²). Correct: d/dx (arctan(5x)) = 5/(1 + (5x)²).
Mistake 2
Ignoring Constants
Students might neglect constants when differentiating. Remember that constants must be included in the differentiation process. For example, differentiate 5 in arctan(5x) to avoid mistakes.
Mistake 3
Not Considering the Domain
The derivative is valid for all real x. However, it's crucial to understand the domain of the original function, arctan(5x), which is (-∞, ∞).
Mistake 4
Confusing arctan(x) and arctan(5x)
Make sure to differentiate arctan(5x) and not confuse it with the derivative of arctan(x), which is 1/(1 + x²). The factor of 5 changes the result.
Examples Using the Derivative of arctan(5x)
Problem 1
Calculate the derivative of arctan(5x)·x².
We have f(x) = arctan(5x)·x². Using the product rule, f'(x) = u′v + uv′. Let u = arctan(5x) and v = x².
Differentiate each term: u′ = 5/(1 + (5x)²) v′ = 2x
Substitute into the product rule: f'(x) = (5/(1 + (5x)²))·x² + arctan(5x)·2x
Thus, the derivative of the function is (5x²)/(1 + (5x)²) + 2x arctan(5x).
Explanation
We differentiate the function by dividing it into two parts and applying the product rule. This involves differentiating each part separately and combining the results.
Problem 2
A car's acceleration is modeled by a = arctan(5x). Find the rate of change of acceleration when x = 1.
We have a = arctan(5x) (acceleration model)...(1)
Differentiate equation (1) to find the rate of change of acceleration: da/dx = 5/(1 + (5x)²)
Substitute x = 1 into the derivative: da/dx = 5/(1 + (5×1)²) = 5/26
Hence, the rate of change of acceleration when x = 1 is 5/26.
Explanation
By differentiating the acceleration function, we find the rate at which acceleration changes at a specific point, x = 1, giving us a value of 5/26.
Problem 3
Derive the second derivative of the function y = arctan(5x).
Find the first derivative: dy/dx = 5/(1 + (5x)²)...(1)
Differentiate equation (1) for the second derivative: d²y/dx² = d/dx [5/(1 + (5x)²)]
Apply the quotient rule: d²y/dx² = -50x/(1 + (5x)²)²
Therefore, the second derivative of y = arctan(5x) is -50x/(1 + (5x)²)².
Explanation
Using the quotient rule, we differentiate the first derivative to obtain the second derivative, simplifying the result to find the final answer.
Problem 4
Prove: d/dx (arctan²(5x)) = 10x arctan(5x)/(1 + (5x)²).
Use the chain rule: Let y = arctan²(5x) = [arctan(5x)]²
Differentiate using the chain rule: dy/dx = 2[arctan(5x)]·d/dx[arctan(5x)] With d/dx[arctan(5x)] = 5/(1 + (5x)²), dy/dx = 2[arctan(5x)]·5/(1 + (5x)²)
Simplify: dy/dx = 10x arctan(5x)/(1 + (5x)²) Hence, proved.
Explanation
Using the chain rule, we differentiate the square of arctan(5x) and substitute the derivative of arctan(5x) to prove the equation.
Problem 5
Solve: d/dx (arctan(5x)/x).
Differentiate using the quotient rule: d/dx (arctan(5x)/x) = (d/dx (arctan(5x))·x - arctan(5x)·d/dx(x))/x²
Substitute d/dx (arctan(5x)) = 5/(1 + (5x)²) and d/dx(x) = 1: = (5x/(1 + (5x)²) - arctan(5x))/x² = (5x - arctan(5x)(1 + (5x)²))/(x²(1 + (5x)²))
Therefore, d/dx (arctan(5x)/x) = (5x - arctan(5x)(1 + (5x)²))/(x²(1 + (5x)²)).
Explanation
Using the quotient rule, we differentiate arctan(5x)/x, substituting the derivatives and simplifying to obtain the final result.
FAQs on the Derivative of arctan(5x)
1.Find the derivative of arctan(5x).
2.Can we use the derivative of arctan(5x) in real life?
3.Is it possible to take the derivative of arctan(5x) at any point?
4.What rule is used to differentiate arctan(5x)/x?
5.How does the derivative of arctan(5x) compare to arctan(x)?
6.Can we find the derivative of the arctan(5x) formula?
Important Glossaries for the Derivative of arctan(5x)
- Derivative: Indicates how a function changes with respect to a variable.
- Arctangent Function: The inverse of the tangent function, denoted as arctan(x).
- Chain Rule: A rule for differentiating composite functions.
- Quotient Rule: A rule used to differentiate ratios of functions.
- Second Derivative: The derivative of the derivative, showing how the rate of change itself changes.
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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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