Domanda del libro di testo
Evaluate the integrals in Exercises 31–78.
33. ∫e^x sec²(e^x - 7)dx
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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.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
Guida verificata passo dopo passo1
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.
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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.
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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.
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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.
Video consigliato:
07:26Derivatives of Inverse Sine & Inverse Cosine
Pratica correlata
Domanda del libro di testo
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
3. y = (1/4)xe^(4x) - (1/16)e^(4x)
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Domanda del libro di testo
Evaluate the integrals in Exercises 31–78.
41. ∫(from 0 to 4)2t/(t² - 25)dt
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Find the volumes of the solids in Exercises 135 and 136.
135. The solid lies between planes perpendicular to the x-axis at x=-1 and x=1. The cross-sections perpendicular to the x-axis are
a. circles whose diameters stretch from the curve y=-1/√(1+x²) to the curve y=1/√(1+x²).
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23. What roles do the functions e^x and ln(x) play in growth comparisons?
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10. True, or false? As x→∞,
a. 1/(x+3) = O(1/x)
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