Textbook Question
Evaluate the integrals in Exercises 31–78.
55. ∫(from -2 to -1)e^(-(x+1)) dx
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- 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.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
5mKey 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
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
19. y = t arctan(t) - 1/2 ln(t)
Textbook Question
Evaluate the integrals in Exercises 31–78.
43. ∫tan(ln v)/v dv
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
In Exercises 115 and 116, find the absolute maximum and minimum values of each function on the given interval.
116. y = 10x (2 - ln(x)), (0, e²]"133. Find the absolute maximum value of
f(x) = x^2 * ln(1/x)
and say where it is assumed
Textbook Question
Evaluate the integrals in Exercises 31–78.
58. ∫(from 0 to ln9)e^θ(e^θ-1)^(1/2) dθ