Textbook Question
Evaluate the integrals in Exercises 39–56.
49. ∫3sec²t/(6 + 3tan(t)) dt
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.6.39
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
39. y=arctan√(x²-1) + arccsc(x), x>1
Verified step by step guidance1
Identify the function to differentiate: \(y = \arctan\left(\sqrt{x^{2} - 1}\right) + \arccsc(x)\), where \(x > 1\).
Recall the derivative formulas: For \(y = \arctan(u)\), \(\frac{dy}{dx} = \frac{1}{1 + u^{2}} \cdot \frac{du}{dx}\); for \(y = \arccsc(x)\), \(\frac{dy}{dx} = -\frac{1}{|x| \sqrt{x^{2} - 1}}\).
Differentiate the first term: Let \(u = \sqrt{x^{2} - 1} = (x^{2} - 1)^{1/2}\). Compute \(\frac{du}{dx}\) using the chain rule: \(\frac{du}{dx} = \frac{1}{2}(x^{2} - 1)^{-1/2} \cdot 2x = \frac{x}{\sqrt{x^{2} - 1}}\).
Apply the derivative formula for \(\arctan(u)\): \(\frac{d}{dx} \arctan\left(\sqrt{x^{2} - 1}\right) = \frac{1}{1 + (\sqrt{x^{2} - 1})^{2}} \cdot \frac{x}{\sqrt{x^{2} - 1}}\).
Differentiate the second term \(\arccsc(x)\) using its derivative formula: \(\frac{d}{dx} \arccsc(x) = -\frac{1}{|x| \sqrt{x^{2} - 1}}\). Since \(x > 1\), \(|x| = x\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative of Inverse Trigonometric Functions
Inverse trigonometric functions like arctan and arccsc have specific derivative formulas. For example, the derivative of arctan(u) is u' / (1 + u²), and the derivative of arccsc(x) is -1 / (|x|√(x² - 1)). Knowing these formulas is essential to differentiate the given function correctly.
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06:35
Derivatives of Other Inverse Trigonometric Functions
Chain Rule
The chain rule is used to differentiate composite functions. When a function is composed of another function, such as arctan(√(x² - 1)), you first differentiate the outer function and then multiply by the derivative of the inner function. This rule is crucial for handling nested expressions.
Recommended video:
05:02Intro to the Chain Rule
Simplification of Derivatives Involving Radicals and Algebraic Expressions
After applying differentiation rules, simplifying expressions involving radicals and algebraic terms like √(x² - 1) is important. This often involves rationalizing denominators or combining terms to present the derivative in a clear, simplified form.
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03:12 Integrals Involving Natural Logs: Substitution Example 8
Related Practice
Textbook Question
In Exercises 73 and 74, repeat the steps above to solve for the functions y=f(x) and x=f^(-1)(y) defined implicitly by the given equations over the interval.
73. y^(1/3) - 1 = (x+2)³, -5 ≤ x ≤ 5, x_0 = -3/2
Textbook Question
Evaluate the integrals in Exercises 53–76.
73. ∫(from 0 to ln√3) e^x dx/(1+e^(2x))
Textbook Question
86. Use a derivative to show that g(x)=√(x² + ln x) is one-to-one.
Textbook Question
In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
70. y = ∛(x(x+1)(x-2)/(x²+1)(2x+3))
Textbook Question
81. Find the lengths of the following curves.
a. y = (x²/8) - ln(x), 4≤x≤8
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