Textbook Question
Each of Exercises 19–24 gives a formula for a function y=f(x) and shows the graphs of f and f^(-1). Find a formula for f^(-1) in each case.
f(x)=x²+1, x≥0
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 7.6.43
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
43. y=√(arcsin x)
Verified step by step guidance1
Identify the function given: \(y = \sqrt{\arcsin x}\). This can be rewritten as \(y = (\arcsin x)^{\frac{1}{2}}\) to make differentiation easier.
Apply the chain rule for differentiation. The chain rule states that if \(y = f(g(x))\), then \(\frac{dy}{dx} = f'(g(x)) \cdot g'(x)\). Here, \(f(u) = u^{\frac{1}{2}}\) and \(g(x) = \arcsin x\).
Differentiate the outer function \(f(u) = u^{\frac{1}{2}}\) with respect to \(u\): \(f'(u) = \frac{1}{2} u^{-\frac{1}{2}}\).
Differentiate the inner function \(g(x) = \arcsin x\) with respect to \(x\): \(g'(x) = \frac{1}{\sqrt{1 - x^2}}\).
Combine the results using the chain rule: \(\frac{dy}{dx} = \frac{1}{2} (\arcsin x)^{-\frac{1}{2}} \cdot \frac{1}{\sqrt{1 - x^2}}\).
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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
The derivative of inverse trigonometric functions like arcsin(x) is essential for this problem. Specifically, the derivative of arcsin(x) with respect to x is 1 / √(1 - x²), which helps in differentiating composite functions involving arcsin.
Recommended video:
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Derivatives of Other Inverse Trigonometric Functions
Chain Rule
The chain rule is used to differentiate composite functions, such as y = √(arcsin x). 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 Square Root Functions
The derivative of a square root function, like √u, can be found using the power rule by rewriting it as u^(1/2). Its derivative is (1/2)u^(-1/2) times the derivative of u, which is crucial when differentiating y = √(arcsin x).
Recommended video:
07:15Root Test
Related Practice
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