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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 7.2.33
Chapter 7, Problem 7.2.33

In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
33. y = ln(sec(lnθ))

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Identify the outer function and the inner functions in the composition. Here, the outer function is the natural logarithm \(y = \ln(u)\) where \(u = \sec(v)\), and the inner function is \(v = \ln\theta\).
Apply the chain rule for derivatives: \(\frac{dy}{d\theta} = \frac{dy}{du} \cdot \frac{du}{dv} \cdot \frac{dv}{d\theta}\).
Compute each derivative separately: \(\frac{dy}{du} = \frac{1}{u}\), \(\frac{du}{dv} = \sec(v) \tan(v)\) (since the derivative of \(\sec x\) is \(\sec x \tan x\)), and \(\frac{dv}{d\theta} = \frac{1}{\theta}\) (derivative of \(\ln \theta\)).
Substitute back the expressions for \(u\) and \(v\) to write the derivative in terms of \(\theta\): \(\frac{dy}{d\theta} = \frac{1}{\sec(\ln \theta)} \cdot \sec(\ln \theta) \tan(\ln \theta) \cdot \frac{1}{\theta}\).
Simplify the expression by canceling terms where possible to get the final derivative formula in terms of \(\theta\).

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Key Concepts

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

Chain Rule

The chain rule is a fundamental differentiation technique used when dealing with composite functions. 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. For example, if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).
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Intro to the Chain Rule

Derivative of Logarithmic Functions

The derivative of the natural logarithm function ln(u) with respect to its variable is 1/u times the derivative of u. Specifically, if y = ln(u), then dy/dx = (1/u) * du/dx. This rule is essential when differentiating functions involving logarithms.
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Derivative of the Natural Logarithmic Function

Derivative of Trigonometric Functions

Knowing the derivatives of trigonometric functions like secant is crucial. The derivative of sec(x) is sec(x)tan(x). When the argument is a function of the variable, the chain rule applies, multiplying by the derivative of the inner function.
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Derivatives of Other Inverse Trigonometric Functions
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