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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 7 - Transcendental FunctionsProblema 7.2.27
Capitolo 7, Problema 7.2.27

In Exercises 7–38, find the derivative of y with respect to x, t, or θ, as appropriate.
27. y = θ(sin(lnθ) + cos(lnθ))

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Identify the function given: \(y = \theta \left( \sin(\ln \theta) + \cos(\ln \theta) \right)\). We need to find \(\frac{dy}{d\theta}\) since the variable is \(\theta\).
Recognize that \(y\) is a product of two functions of \(\theta\): \(u = \theta\) and \(v = \sin(\ln \theta) + \cos(\ln \theta)\). We will use the product rule for differentiation: \(\frac{d}{d\theta}(uv) = u'v + uv'\).
Compute the derivative of \(u = \theta\), which is \(u' = 1\).
Compute the derivative of \(v = \sin(\ln \theta) + \cos(\ln \theta)\). Use the chain rule for each term: For \(\sin(\ln \theta)\), derivative is \(\cos(\ln \theta) \cdot \frac{1}{\theta}\); for \(\cos(\ln \theta)\), derivative is \(-\sin(\ln \theta) \cdot \frac{1}{\theta}\). Then sum these results to find \(v'\).
Apply the product rule: \(\frac{dy}{d\theta} = u'v + uv' = 1 \cdot \left( \sin(\ln \theta) + \cos(\ln \theta) \right) + \theta \cdot \left( \cos(\ln \theta) \cdot \frac{1}{\theta} - \sin(\ln \theta) \cdot \frac{1}{\theta} \right)\). Simplify the expression as much as possible.

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Chain Rule

The chain rule is used to differentiate composite functions. When a function is nested inside another, such as sin(lnθ), you differentiate the outer function first and multiply by the derivative of the inner function. This rule is essential for handling functions like sin(lnθ) and cos(lnθ).
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Intro to the Chain Rule

Product Rule

The product rule is applied when differentiating the product of two functions, such as θ multiplied by (sin(lnθ) + cos(lnθ)). It states that the derivative of f(θ)g(θ) is f'(θ)g(θ) + f(θ)g'(θ), allowing you to differentiate each part separately and combine the results.
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The Product Rule

Derivatives of Logarithmic and Trigonometric Functions

Knowing the derivatives of lnθ, sin(x), and cos(x) is crucial. The derivative of lnθ is 1/θ, while the derivatives of sin(x) and cos(x) are cos(x) and -sin(x), respectively. These rules help differentiate the inner and outer functions in the expression.
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Derivative of the Natural Logarithmic Function
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