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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.7.19
Chapter 7, Problem 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))

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1
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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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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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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The 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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Intro to the Chain Rule
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