Domanda del libro di testo
In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
64. y = 1/(t(t+1)(t+2))
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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 7.6.29
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
29. y=arcsec(1/t), 0<t<1
Guida verificata passo dopo passo1
Identify the function given: \(y = \arcsec\left(\frac{1}{t}\right)\), where \(0 < t < 1\).
Recall the derivative formula for \(y = \arcsec(u)\) with respect to \(t\):
\(\frac{dy}{dt} = \frac{1}{|u| \sqrt{u^2 - 1}} \cdot \frac{du}{dt}\), where \(u\) is a function of \(t\).
Set \(u = \frac{1}{t}\) and compute its derivative with respect to \(t\):
\(\frac{du}{dt} = \frac{d}{dt} \left(\frac{1}{t}\right) = -\frac{1}{t^2}\).
Substitute \(u\) and \(\frac{du}{dt}\) back into the derivative formula:
\(\frac{dy}{dt} = \frac{1}{\left| \frac{1}{t} \right| \sqrt{\left(\frac{1}{t}\right)^2 - 1}} \cdot \left(-\frac{1}{t^2}\right)\).
Simplify the expression step-by-step, paying attention to the absolute value and the square root, to express \(\frac{dy}{dt}\) in terms of \(t\).
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Derivative of Inverse Trigonometric Functions
Inverse trigonometric functions like arcsec(x) have specific derivative formulas. For arcsec(x), the derivative is 1 / (|x|√(x² - 1)). Understanding these formulas is essential to differentiate functions involving inverse trig expressions.
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06:35
Derivatives of Other Inverse Trigonometric Functions
Chain Rule
The chain rule is used to differentiate composite functions. When y = arcsec(1/t), you first differentiate arcsec(u) with respect to u, then multiply by the derivative of u = 1/t. This rule allows handling nested functions systematically.
Video consigliato:
05:02Intro to the Chain Rule
Domain Considerations for arcsec(x)
The domain of arcsec(x) is |x| ≥ 1, but here the argument is 1/t with 0 < t < 1, so 1/t > 1, which fits the domain. Recognizing domain restrictions ensures the function and its derivative are valid in the given interval.
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Percorso guidato
5:10Finding the Domain and Range of a Graph
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