Domanda del libro di testo
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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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 7.6.37
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
37. y=s√(1-s²) + arccos(s)
Guida verificata passo dopo passo1
Identify the function to differentiate: \(y = s \sqrt{1 - s^2} + \arccos(s)\), where \(y\) is expressed in terms of \(s\).
Rewrite the square root term for easier differentiation: \(s \sqrt{1 - s^2} = s (1 - s^2)^{1/2}\).
Apply the product rule to differentiate \(s (1 - s^2)^{1/2}\). Recall the product rule: \(\frac{d}{ds}[u v] = u' v + u v'\), where \(u = s\) and \(v = (1 - s^2)^{1/2}\).
Differentiate \(\arccos(s)\) using the chain rule. The derivative of \(\arccos(s)\) with respect to \(s\) is \(-\frac{1}{\sqrt{1 - s^2}}\).
Combine the derivatives from the product rule and the derivative of \(\arccos(s)\) to write the full expression for \(\frac{dy}{ds}\).
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Derivative of Composite Functions
This involves applying the chain rule when differentiating functions composed of other functions, such as √(1 - s²). The chain rule 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:
3:48
Evaluate Composite Functions - Special Cases
Derivative of Inverse Trigonometric Functions
The derivative of arccos(s) with respect to s is -1/√(1 - s²). Understanding the derivatives of inverse trig functions is essential for differentiating expressions involving arccos or arcsin.
Video consigliato:
06:35Derivatives of Other Inverse Trigonometric Functions
Product Rule for Differentiation
When differentiating a product of two functions, such as s and √(1 - s²), the product rule is used. It states that the derivative of f(s)g(s) is f'(s)g(s) + f(s)g'(s), combining the derivatives of each function appropriately.
Video consigliato:
05:18The Product Rule
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