Domanda del libro di testo
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
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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
Capitolo 7, Problema 7.6.43
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
43. y=√(arcsin x)
Guida verificata passo dopo passo1
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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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.
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06:35
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.
Video consigliato:
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).
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Percorso guidato
07:15Root Test
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