Domanda del libro di testo
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
14. lim (t → 0) sin 5t / 2t
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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 7.7.19
In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
19. y = (sech θ)(1-ln(sech θ))
Guida verificata passo dopo passo1
Identify the function to differentiate: \(y = (\text{sech} \ \theta)(1 - \ln(\text{sech} \ \theta))\). Notice this is a product of two functions of \(\theta\).
Apply the product rule for derivatives: If \(y = u(\theta) \cdot v(\theta)\), then \(\frac{dy}{d\theta} = u'(\theta) v(\theta) + u(\theta) v'(\theta)\), where \(u(\theta) = \text{sech} \ \theta\) and \(v(\theta) = 1 - \ln(\text{sech} \ \theta)\).
Find the derivative of \(u(\theta) = \text{sech} \ \theta\). Recall that \(\frac{d}{d\theta} \text{sech} \ \theta = -\text{sech} \ \theta \tanh \ \theta\).
Find the derivative of \(v(\theta) = 1 - \ln(\text{sech} \ \theta)\). Use the chain rule: \(\frac{d}{d\theta} \ln(\text{sech} \ \theta) = \frac{1}{\text{sech} \ \theta} \cdot \frac{d}{d\theta} \text{sech} \ \theta\). Substitute the derivative of \(\text{sech} \ \theta\) from the previous step.
Combine all parts using the product rule formula: \(\frac{dy}{d\theta} = u'(\theta) v(\theta) + u(\theta) v'(\theta)\), then simplify the expression as much as possible.
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Derivative of Hyperbolic Functions
Hyperbolic functions like sech(θ) have specific derivatives; for example, the derivative of sech(θ) is -sech(θ) tanh(θ). Understanding these derivatives is essential to differentiate expressions involving hyperbolic functions correctly.
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Percorso guidato
5:50
Asymptotes of Hyperbolas
Product Rule
The product rule is used to differentiate the product of two functions. It states that (fg)' = f'g + fg', meaning you differentiate each function separately and combine the results appropriately.
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05:18The Product Rule
Chain Rule
The chain rule helps differentiate composite functions, such as ln(sech(θ)). It requires differentiating the outer function and multiplying by the derivative of the inner function, enabling correct handling of nested expressions.
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05:02Intro to the Chain Rule
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