Textbook Question
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
19. y = t arctan(t) - 1/2 ln(t)
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.PE.15
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
15. y = sin⁻¹√(1-u²), 0<u<1
Verified step by step guidance1
Identify the function given: \(y = \sin^{-1}\left(\sqrt{1 - u^2}\right)\), where \(0 < u < 1\).
Recall the derivative formula for the inverse sine function: if \(y = \sin^{-1}(x)\), then \(\frac{dy}{dx} = \frac{1}{\sqrt{1 - x^2}}\).
Set the inner function as \(g(u) = \sqrt{1 - u^2}\) and apply the chain rule: \(\frac{dy}{du} = \frac{1}{\sqrt{1 - (g(u))^2}} \cdot g'(u)\).
Calculate \(g'(u)\) by differentiating \(g(u) = (1 - u^2)^{1/2}\) using the power rule and chain rule: \(g'(u) = \frac{1}{2}(1 - u^2)^{-1/2} \cdot (-2u)\).
Substitute \(g(u)\) and \(g'(u)\) back into the expression for \(\frac{dy}{du}\) and simplify the resulting expression step-by-step.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Trigonometric Functions
Inverse trigonometric functions, like sin⁻¹(x), are the inverses of the standard trig functions and return an angle whose trigonometric value is x. Understanding their domains and ranges is essential for correctly differentiating expressions involving them.
Recommended video:
06:35
Derivatives of Other Inverse Trigonometric Functions
Chain Rule
The chain rule is a differentiation technique used when a function is composed of another function. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function.
Recommended video:
05:02Intro to the Chain Rule
Derivative of Inverse Sine Function
The derivative of y = sin⁻¹(x) with respect to x is 1/√(1 - x²), valid for |x| < 1. This formula is crucial when differentiating expressions involving inverse sine, especially when combined with the chain rule.
Recommended video:
07:26Derivatives of Inverse Sine & Inverse Cosine
Related Practice
Textbook Question
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
93. lim(x→0) (csc(x) - cot(x))
Textbook Question
110. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
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Textbook Question
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
104. lim(x→4) (sin²(πx))/(e^(x-4) + 3 - x)
Textbook Question
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
23. y = arccsc(secθ), 0<θ<π/2
Textbook Question
Evaluate the integrals in Exercises 31–78.
58. ∫(from 0 to ln9)e^θ(e^θ-1)^(1/2) dθ