Textbook Question
Evaluate the integrals in Exercises 111–114.
111. ∫₁^(ln x) (1 / t) dt,x > 1
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 7.6.25
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
25. y=arcsec(2s+1)
Verified step by step guidance1
Identify the function given: \(y = \arcsec(2s + 1)\). We need to find \(\frac{dy}{ds}\), the derivative of \(y\) with respect to \(s\).
Recall the derivative formula for \(y = \arcsec(u)\), where \(u\) is a function of \(s\):
\(\frac{dy}{ds} = \frac{1}{|u| \sqrt{u^2 - 1}} \cdot \frac{du}{ds}\).
Set \(u = 2s + 1\). Compute the derivative of \(u\) with respect to \(s\):
\(\frac{du}{ds} = 2\).
Substitute \(u\) and \(\frac{du}{ds}\) into the derivative formula:
\(\frac{dy}{ds} = \frac{1}{|2s + 1| \sqrt{(2s + 1)^2 - 1}} \cdot 2\).
Simplify the expression inside the square root and write the final derivative expression in terms of \(s\), leaving it in simplified radical form without calculating the numeric value.
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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 arcsec(x) have specific derivative formulas. Understanding these formulas is essential to differentiate functions involving inverse trig expressions correctly.
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Derivatives of Other Inverse Trigonometric Functions
Chain Rule
The chain rule is used to differentiate composite functions. When the argument of arcsec is a function of x, such as 2s+1, the derivative of the outer function is multiplied by the derivative of the inner function.
Recommended video:
05:02Intro to the Chain Rule
Domain and Differentiability of arcsec(x)
The function arcsec(x) is defined for |x| ≥ 1, and its derivative involves expressions with sqrt(x^2 - 1). Recognizing the domain restrictions and the form of the derivative helps avoid errors in differentiation.
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5:10Finding the Domain and Range of a Graph
Related Practice
Textbook Question
Solve the initial value problems in Exercises 55–58.
57. d²y/dx² = 2e^(−x),y(0) = 1,y′(0) = 0
Textbook Question
In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
66. y = θsin(θ)/√(sec(θ))
Textbook Question
Evaluate the integrals in Exercises 77–90.
77. ∫dx/√(-x²+4x-3)
Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
22. lim (x → 1) (x - 1) / (ln x - sin πx)
Textbook Question
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = e^(θ)(sinθ + cosθ)