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103 LearnersLast updated on September 15, 2025
Derivative of Tan^3(x)
We explore the derivative of tan³(x), which involves using the chain rule and the power rule to understand how the function changes with respect to x. Derivatives are crucial for analyzing changes and trends in various fields. This section will delve into the derivative of tan³(x) in detail.
What is the Derivative of Tan^3(x)?
We will explore the derivative of tan³(x). It is expressed as d/dx (tan³(x)) or (tan³(x))', and its derivative involves applying the chain rule and power rule. The derivative signifies the rate of change of the function tan³(x) within its domain. Key concepts include:
Tangent Function: tan(x) = sin(x)/cos(x).
Power Rule: Used for differentiating functions raised to a power.
Chain Rule: Applied for differentiating composite functions.
Derivative of Tan^3(x) Formula
The derivative of tan³(x) can be denoted as d/dx (tan³(x)) or (tan³(x))'.
Using the chain rule and power rule, the formula is: d/dx (tan³(x)) = 3 tan²(x) sec²(x).
This formula is valid for all x where cos(x) ≠ 0.
Proofs of the Derivative of Tan^3(x)
We can derive the derivative of tan³(x) using various methods. The proofs involve trigonometric identities and differentiation rules. Here are some methods used to derive this:
- Using Chain Rule
- Using First Principle
- Using Product Rule
Let's demonstrate the derivation of the derivative of tan³(x) as 3 tan²(x) sec²(x) using these methods:
Using Chain Rule
To differentiate tan³(x) using the chain rule: Consider y = (tan(x))³
We apply the chain rule: dy/dx = 3 (tan(x))² * d/dx(tan(x))
Since d/dx(tan(x)) = sec²(x), we get: dy/dx = 3 tan²(x) sec²(x)
Using First Principle
The first principle can also be applied, but it involves more complex computations and is generally not preferred for composite functions like tan³(x).
Using Product Rule
Express tan³(x) as (tan(x))(tan(x))(tan(x)) and apply the product rule iteratively, which will ultimately result in the same derivative: 3 tan²(x) sec²(x).
Higher-Order Derivatives of Tan^3(x)
Higher-order derivatives are obtained by differentiating a function multiple times. They reveal further nuances of how the function behaves.
For tan³(x): The first derivative is f′(x) = 3 tan²(x) sec²(x).
The second derivative is obtained by differentiating the first derivative.
This process continues to reveal how the rate of change itself changes, offering deeper insights into the function's behavior.
Special Cases:
When x is π/2, the derivative is undefined due to the vertical asymptote of tan(x). When x is 0, the derivative of tan³(x) is 0, as tan(0) = 0.
Common Mistakes and How to Avoid Them in Derivatives of Tan^3(x)
Differentiating tan³(x) can lead to errors if the rules are not applied correctly. Here are common mistakes and tips to avoid them:
Mistake 1
Ignoring the Chain Rule
It's crucial to apply the chain rule when differentiating tan³(x). Failing to do so, such as directly applying the power rule, will yield incorrect results. Always remember to differentiate the inner function tan(x) before applying the power rule.
Mistake 2
Forgetting the Undefined Points of Tan x
Students often overlook the undefined points of tan(x), such as x = π/2. Always consider the domain where the function is continuous and differentiable.
Mistake 3
Misapplying the Power Rule
When applying the power rule, students sometimes forget to multiply by the derivative of the inner function. For tan³(x), remember to use both the power rule and the chain rule.
Mistake 4
Incorrectly Simplifying the Expression
Ensure proper simplification of expressions. Mistakes often occur when simplifying terms like tan²(x) sec²(x). Double-check each step for accuracy.
Mistake 5
Neglecting Constants and Coefficients
When differentiating expressions like 5 tan³(x), it's important to multiply by the constant. For example, d/dx (5 tan³(x)) = 5 * 3 tan²(x) sec²(x).
Examples Using the Derivative of Tan^3(x)
Problem 1
Calculate the derivative of tan³(x)·sec²(x)
Let f(x) = tan³(x)·sec²(x). Using the product rule, f'(x) = u′v + uv′
Here, u = tan³(x) and v = sec²(x).
Differentiate each term: u′ = d/dx (tan³(x)) = 3 tan²(x) sec²(x) v′ = d/dx (sec²(x)) = 2 sec²(x) tan(x)
Substitute into the equation: f'(x) = (3 tan²(x) sec²(x))·(sec²(x)) + (tan³(x))·(2 sec²(x) tan(x))
Simplify to get: f'(x) = 3 tan²(x) sec⁴(x) + 2 tan⁴(x) sec²(x)
Explanation
We find the derivative by using the product rule. First, we differentiate each part separately and then combine them to get the final result.
Problem 2
XYZ Corporation is analyzing a cost function modeled by y = tan³(x) to predict production costs. If x = π/6, calculate the rate of cost change.
Given y = tan³(x), Differentiate: dy/dx = 3 tan²(x) sec²(x)
Substitute x = π/6: tan(π/6) = 1/√3 and sec(π/6) = 2/√3 dy/dx = 3 (1/√3)² (2/√3)² = 3 (1/3) (4/9) = 4/9
The rate of cost change at x = π/6 is 4/9.
Explanation
We substitute x = π/6 into the derivative formula to find how the cost function changes at that point. Simplifying the trigonometric values gives us the rate of change.
Problem 3
Derive the second derivative of y = tan³(x).
First, find the first derivative: dy/dx = 3 tan²(x) sec²(x)
Now, find the second derivative: d²y/dx² = d/dx [3 tan²(x) sec²(x)]
Use the product rule: d²y/dx² = 3 [2 tan(x) sec²(x) sec²(x) + tan²(x) (2 sec²(x) tan(x))] = 6 tan(x) sec⁴(x) + 6 tan³(x) sec²(x)
Therefore, the second derivative is 6 tan(x) sec⁴(x) + 6 tan³(x) sec²(x).
Explanation
We differentiate the first derivative using the product rule, taking care to apply the chain rule where necessary. This gives the second derivative of the function.
Problem 4
Prove: d/dx (tan²(x)) = 2 tan(x) sec²(x).
Start with y = tan²(x). Express as [tan(x)]².
Differentiate 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)
Thus, d/dx(tan²(x)) = 2 tan(x) sec²(x).
Explanation
We used the chain rule to differentiate tan²(x), substituting the derivative of tan(x) into the equation, resulting in the proven formula.
Problem 5
Solve: d/dx (tan³(x)/x)
Use the quotient rule: d/dx (tan³(x)/x) = (d/dx (tan³(x))·x - tan³(x)·d/dx(x))/x²
Substitute: d/dx(tan³(x)) = 3 tan²(x) sec²(x) and d/dx(x) = 1 = [3 tan²(x) sec²(x)·x - tan³(x)·1]/x² = [3x tan²(x) sec²(x) - tan³(x)]/x²
Therefore, d/dx (tan³(x)/x) = [3 tan²(x) sec²(x) - tan³(x)/x].
Explanation
We apply the quotient rule to differentiate the given function, carefully simplifying each term to reach the final solution.
FAQs on the Derivative of Tan^3(x)
1.Find the derivative of tan³(x).
2.Can the derivative of tan³(x) be applied in real life?
3.Is it possible to differentiate tan³(x) at x = π/2?
4.What rule is used to differentiate tan³(x)/x?
5.Are the derivatives of tan³(x) and tan⁻¹(x) the same?
Important Glossaries for the Derivative of Tan^3(x)
- Derivative: Indicates how a function changes with respect to a variable.
- Tangent Function: A trigonometric function represented as tan(x).
- Secant Function: The reciprocal of the cosine function, denoted as sec(x).
- Chain Rule: A differentiation rule for composite functions.
- Power Rule: A differentiation rule for functions raised to a power.
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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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