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106 LearnersLast updated on October 8, 2025
Derivative of Tan²x
We explore the derivative of tan²(x), which is 2 tan(x) sec²(x), as a tool for understanding how the function tan²(x) changes concerning changes in x. Derivatives are crucial in calculating various changes in real-life situations. Let's delve into the details of the derivative of tan²(x).
What is the Derivative of Tan²x?
The derivative of tan²x can be represented as d/dx (tan²x) or (tan²x)', and its value is 2 tan(x) sec²(x). The function tan²x is differentiable within its domain.
Key concepts include:
Tangent Function: tan(x) = sin(x)/cos(x).
Chain Rule: Used for differentiating composite functions like tan²(x).
Secant Function: sec(x) = 1/cos(x).
Derivative of Tan²x Formula
The derivative of tan²x is denoted as d/dx (tan²x) or (tan²x)'.
The formula is: d/dx (tan²x) = 2 tan(x) sec²(x)
This formula is applicable for all x where cos(x) ≠ 0.
Proofs of the Derivative of Tan²x
To derive the derivative of tan²x, we use various methods of differentiation, including:
- By Chain Rule
- By Product Rule
By Chain Rule
Let y = tan²(x) = (tan(x))². Differentiating using the chain rule: dy/dx = 2 tan(x) d/dx (tan(x)) Since d/dx (tan(x)) = sec²(x), dy/dx = 2 tan(x) sec²(x). Hence, proved.
Using Product Rule
Express tan²(x) as (tan(x))(tan(x)) and use the product rule: d/dx (tan²x) = d/dx [(tan(x))(tan(x))] = tan(x) d/dx (tan(x)) + tan(x) d/dx (tan(x)) = tan(x) sec²(x) + tan(x) sec²(x) = 2 tan(x) sec²(x). Hence, proved.
Higher-Order Derivatives of Tan²x
Higher-order derivatives result from differentiating a function multiple times. They provide insights into the function's behavior.
For tan²(x), the first derivative is f′(x), which describes the rate of change, while the second derivative, f′′(x), indicates the rate of that change. This pattern continues for higher derivatives.
Special Cases
When x is π/2, the derivative is undefined due to a vertical asymptote in tan(x).
When x is 0, the derivative of tan²x = 2 tan(0) sec²(0) = 0.
Common Mistakes and How to Avoid Them in Derivatives of Tan²x
Students often make errors when differentiating tan²x. Understanding the correct methods helps avoid these mistakes.
Mistake 1
Not Applying the Chain Rule Correctly
Students may overlook using the chain rule, leading to incorrect results.
Always identify the inner and outer functions and differentiate accordingly.
Mistake 2
Forgetting the Undefined Points of Tan²x
Students might forget that tan²x is undefined at x = π/2, 3π/2,...
Always consider the domain of the function.
Mistake 3
Incorrect Use of Product Rule
Errors occur when misapplying the product rule.
For instance, incorrect differentiation of tan(x)tan(x). Always apply the rule correctly and simplify.
Mistake 4
Neglecting to Multiply Constants
Sometimes constants are ignored in differentiation, like writing d/dx (5 tan²x) = 5 sec²x.
Ensure all constants are multiplied correctly.
Mistake 5
Mixing Up Derivatives of tan²x and tan x
The derivative of tan²x is not the same as tan x.
Ensure you differentiate tan²x using the correct methods.
Examples Using the Derivative of Tan²x
Problem 1
Calculate the derivative of (tan²x · sec²x).
For f(x) = tan²x · sec²x, use the product rule: f'(x) = u′v + uv′ Where u = tan²x and v = sec²x. u′ = d/dx (tan²x) = 2 tan(x) sec²(x) v′ = d/dx (sec²x) = 2 sec²x tan(x) Substituting: f'(x) = (2 tan(x) sec²(x)) sec²x + tan²x (2 sec²x tan(x)) = 2 tan(x) sec⁴x + 2 tan³x sec²x.
Explanation
Differentiate the given function using the product rule.
Break it into two parts, find their derivatives, and combine them.
Problem 2
The elevation of a hill is represented by y = tan²(x) where y is the height at a distance x. If x = π/6, find the slope of the hill.
Given y = tan²(x), differentiate: dy/dx = 2 tan(x) sec²(x) Substitute x = π/6: dy/dx = 2 tan(π/6) sec²(π/6) tan(π/6) = 1/√3, sec(π/6) = 2/√3 dy/dx = 2(1/√3)(4/3) = 8/9.
Explanation
Find the slope by differentiating y = tan²(x) and substituting x = π/6.
Calculate the result using trigonometric values.
Problem 3
Derive the second derivative of the function y = tan²(x).
First derivative: dy/dx = 2 tan(x) sec²(x). Now, differentiate again: d²y/dx² = d/dx [2 tan(x) sec²(x)] Use the product rule: d²y/dx² = 2[sec²(x) sec²(x) + tan(x) 2 sec²(x) tan(x)] = 2 sec⁴(x) + 4 tan²(x) sec²(x).
Explanation
Use the product rule for the second derivative.
Differentiate the first derivative and simplify.
Problem 4
Prove: d/dx (tan²(x)) = 2 tan(x) sec²(x).
Use the chain rule: y = tan²(x) = (tan(x))² dy/dx = 2 tan(x) d/dx (tan(x)) = 2 tan(x) sec²(x), hence proved.
Explanation
Differentiate using the chain rule by considering the inner and outer functions, then simplify.
Problem 5
Solve: d/dx (tan²x / x).
Use the quotient rule: d/dx (tan²x / x) = (x d/dx (tan²x) - tan²x d/dx (x)) / x² = (x (2 tan(x) sec²(x)) - tan²x) / x² = (2x tan(x) sec²(x) - tan²x) / x².
Explanation
Differentiate using the quotient rule.
Simplify the result by finding the derivatives of the numerator and denominator.
FAQs on the Derivative of Tan²x
1.Find the derivative of tan²x.
2.Can the derivative of tan²x be used in real life?
3.Is it possible to take the derivative of tan²x at x = π/2?
4.What rule is used to differentiate tan²x / x?
5.Are the derivatives of tan²x and tan(x) the same?
Important Glossaries for the Derivative of Tan²x
- Derivative: Measures the rate of change of a function concerning its variable.
- Chain Rule: A rule for differentiating composite functions.
- Tangent Function: A trigonometric function expressed as tan(x) = sin(x)/cos(x).
- Secant Function: A trigonometric function, sec(x) = 1/cos(x).
- Asymptote: A line that a graph approaches but never touches.
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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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: He loves to play the quiz with kids through algebra to make kids love it.
