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104 LearnersLast updated on August 5, 2025
Derivative of 2 sec²x tanx
We use the derivative of 2 sec²x tanx to understand how this composite function changes with respect to x. Derivatives are crucial in various fields such as physics and engineering to understand rates of change. We will explore the derivative of 2 sec²x tanx in detail.
What is the Derivative of 2 sec²x tanx?
The derivative of 2 sec²x tanx can be represented as d/dx (2 sec²x tanx). This function is differentiable within its domain. Here are the key concepts involved: Secant Function: sec(x) = 1/cos(x). Tangent Function: tan(x) = sin(x)/cos(x). Product Rule: Used for differentiating the product of two functions.
Derivative of 2 sec²x tanx Formula
The derivative of 2 sec²x tanx can be found using the product and chain rules. The formula is: d/dx (2 sec²x tanx)
Proofs of the Derivative of 2 sec²x tanx
We can derive the derivative of 2 sec²x tanx by using differentiation rules. Here are some methods we can use: Using Product Rule To differentiate 2 sec²x tanx, let's consider u = 2 sec²x and v = tanx. By the product rule: d/dx [u . v] = u'v + uv', where u' is the derivative of u and v' is the derivative of v. u = 2 sec²x, so u' = 2(2 sec²x tanx) = 4 sec²x tanx (using the chain rule). v = tanx, so v' = sec²x. Thus, d/dx (2 sec²x tanx) = (4 sec²x tanx) tanx + (2 sec²x)(sec²x) = 4 sec²x tan²x + 2 sec⁴x
Higher-Order Derivatives of 2 sec²x tanx
When a function is differentiated multiple times, the resulting derivatives are called higher-order derivatives. For example, the first derivative tells us the rate of change, while the second derivative indicates how this rate changes. Calculating higher-order derivatives of 2 sec²x tanx can reveal more about the function's behavior. The first derivative is indicated as f′(x), showing how the function changes at a given point. The second derivative, f′′(x), is derived from the first and tells us about the acceleration of the rate of change. This pattern continues for higher derivatives.
Special Cases:
When x is π/2, the derivative is undefined because sec(x) and tan(x) have vertical asymptotes there. When x is 0, the derivative of 2 sec²x tanx is 0 because tan(0) = 0.
Common Mistakes and How to Avoid Them in Derivatives of 2 sec²x tanx
Students often make errors when differentiating 2 sec²x tanx. Understanding the correct approach can help avoid these mistakes. Here are some common errors and solutions:
Mistake 1
Forgetting to Apply the Product Rule
Students might forget to use the product rule when differentiating products of functions, leading to incorrect results. Always ensure to apply the product rule step by step.
Mistake 2
Ignoring Undefined Points
Students often overlook that sec(x) and tan(x) are undefined at certain points like (x = π/2, 3π/2,...). Always consider the domain of the function to avoid errors.
Mistake 3
Misapplying the Chain Rule
When differentiating composite functions, students may not apply the chain rule correctly. For example, forgetting the derivative of the inner function. Always identify inner and outer functions and differentiate accordingly.
Mistake 4
Incorrect Coefficient Handling
Students sometimes neglect to multiply constants appropriately. For instance, they might forget to multiply by 2 when differentiating 2 sec²x tanx. Check all coefficients to ensure accuracy.
Mistake 5
Neglecting Higher-Order Derivatives
When asked for higher-order derivatives, students might stop at the first derivative. Remember that higher derivatives provide more insights into the function's behavior.
Examples Using the Derivative of 2 sec²x tanx
Problem 1
Calculate the derivative of (2 sec²x tanx · sinx).
Here, we have f(x) = 2 sec²x tanx · sinx. Using the product rule, f'(x) = u′v + uv′ In the given equation, u = 2 sec²x tanx and v = sinx. Let's differentiate each term, u′ = d/dx (2 sec²x tanx) = 4 sec²x tan²x + 2 sec⁴x v′ = d/dx (sinx) = cosx Substituting these into the equation, f'(x) = (4 sec²x tan²x + 2 sec⁴x) sinx + (2 sec²x tanx) cosx Simplify the expression to get the final answer.
Explanation
We differentiate the function by applying the product rule. First, find the derivatives of each part, then combine them using the product rule to achieve the final result.
Problem 2
A rotating radar dish's angle of elevation is given by θ = 2 sec²x tanx, where x is time. Find the rate of change of the angle when x = π/6.
We have θ = 2 sec²x tanx ... (1) Differentiate equation (1) with respect to x: dθ/dx = 4 sec²x tan²x + 2 sec⁴x Substitute x = π/6 into the derivative: sec(π/6) = 2/√3, tan(π/6) = 1/√3 dθ/dx = 4(2/√3)²(1/√3)² + 2(2/√3)⁴ = 4(4/3)(1/3) + 2(16/9) = 16/9 + 32/9 = 48/9 = 16/3 Thus, the rate of change of the angle is 16/3.
Explanation
We calculate the rate of change of the angle by differentiating the function and substituting x = π/6. After simplification, we find the rate of change, which describes how fast the angle changes at that specific moment.
Problem 3
Derive the second derivative of the function y = 2 sec²x tanx.
First, find the first derivative: dy/dx = 4 sec²x tan²x + 2 sec⁴x ... (1) Differentiate equation (1) to get the second derivative: d²y/dx² = d/dx [4 sec²x tan²x + 2 sec⁴x] Apply the product rule and chain rule: d²y/dx² = 8 sec²x tanx (2 sec²x tanx) + 8 sec²x tan³x + 8 sec⁶x tanx Simplify to find the second derivative.
Explanation
We use the first derivative and apply the product and chain rules again to find the second derivative. This process helps us understand the function's concavity and acceleration.
Problem 4
Prove: d/dx (sec²x) = 2 sec²x tanx.
Consider y = sec²x To differentiate, use the chain rule: dy/dx = 2 sec(x) d/dx [sec(x)] Since d/dx [sec(x)] = sec(x) tan(x), dy/dx = 2 sec(x) sec(x) tan(x) dy/dx = 2 sec²x tanx Hence proved.
Explanation
We use the chain rule to differentiate sec²x. First, differentiate sec(x), then multiply by the derivative of sec(x) to get the final result.
Problem 5
Solve: d/dx (2 sec²x tanx / x)
To differentiate the function, use the quotient rule: d/dx (2 sec²x tanx / x) = (d/dx (2 sec²x tanx) · x - 2 sec²x tanx · d/dx(x)) / x² Substitute d/dx (2 sec²x tanx) = 4 sec²x tan²x + 2 sec⁴x and d/dx (x) = 1: = [(4 sec²x tan²x + 2 sec⁴x) · x - 2 sec²x tanx] / x² Simplify the expression to get the final result.
Explanation
We differentiate the given function using the quotient rule. After calculating derivatives of the numerator and denominator, we simplify to obtain the final answer.
FAQs on the Derivative of 2 sec²x tanx
1.Find the derivative of 2 sec²x tanx.
2.Can the derivative of 2 sec²x tanx be used in real life?
3.Is it possible to take the derivative of 2 sec²x tanx at x = π/2?
4.What rule is used to differentiate 2 sec²x tanx / x?
5.Are the derivatives of sec²x and sec²x tanx the same?
Important Glossaries for the Derivative of 2 sec²x tanx
Derivative: Represents the rate of change of a function with respect to a variable. Product Rule: A differentiation rule used for differentiating products of two functions. Quotient Rule: A differentiation rule used for differentiating quotients of two functions. Chain Rule: A rule for differentiating composite functions by differentiating the outer function and multiplying by the derivative of the inner function. Secant Function: A trigonometric function, sec(x) is the reciprocal of cos(x).
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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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