Textbook Question
Evaluate the integrals in Exercises 53–76.
63. ∫(from -1 to -√2/2)dy/(y√(4y²-1))
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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 7.7.25
In Exercises 25–36, find the derivative of y with respect to the appropriate variable.
25. y = sinh⁻¹(√x)
Verified step by step guidance1
Identify the function: \( y = \sinh^{-1}(\sqrt{x}) \). Here, \( y \) is the inverse hyperbolic sine of \( \sqrt{x} \).
Recall the derivative formula for the inverse hyperbolic sine function: \( \frac{d}{dx} \sinh^{-1}(u) = \frac{1}{\sqrt{1+u^2}} \cdot \frac{du}{dx} \), where \( u \) is a function of \( x \).
Set \( u = \sqrt{x} = x^{1/2} \). Then find \( \frac{du}{dx} \) using the power rule: \( \frac{du}{dx} = \frac{1}{2} x^{-1/2} \).
Apply the chain rule: \( \frac{dy}{dx} = \frac{1}{\sqrt{1 + (\sqrt{x})^2}} \cdot \frac{1}{2} x^{-1/2} \).
Simplify the expression inside the square root: \( (\sqrt{x})^2 = x \), so the derivative becomes \( \frac{dy}{dx} = \frac{1}{\sqrt{1 + x}} \cdot \frac{1}{2} x^{-1/2} \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Inverse Hyperbolic Sine Function (sinh⁻¹)
The inverse hyperbolic sine function, sinh⁻¹(x), is the inverse of the hyperbolic sine function. It can be expressed as sinh⁻¹(x) = ln(x + √(x² + 1)), which helps in differentiating expressions involving sinh⁻¹.
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Chain Rule
The chain rule is a differentiation technique used when a function is composed of other functions. 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.
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Derivative of Square Root Function
The square root function, √x, can be written as x^(1/2). Its derivative is (1/2)x^(-1/2), which is essential when differentiating expressions where the square root is inside another function.
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Related Practice
Textbook Question
Solve the differential equation in Exercises 9–22.
21. (1/x)(dy/dx) = ye^(x²) + 2√y e^(x²)
Textbook Question
In Exercises 7–10, determine from its graph if the function is one-to-one.
f(x) = 3 - x, x < 0
= 3, x ≥ 0
Textbook Question
15. Find f'(2) if f(x) = e^(g(x)) and g(x) = ∫(from 2 to x) t/(1+t⁴)dt.
Textbook Question
In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
122. y = (ln x)^(ln x)
Textbook Question
Find the limits in Exercises 1–6.
5. lim(n→∞) (1/(n+1) + 1/(n+2) + ... + 1/(2n))