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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 7.2.66
Chapter 7, Problem 7.2.66

In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
66. y = θsin(θ)/√(sec(θ))

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Start by rewriting the function to make logarithmic differentiation easier. Given \( y = \frac{\theta \sin(\theta)}{\sqrt{\sec(\theta)}} \), express the denominator with a fractional exponent: \( y = \theta \sin(\theta) \cdot (\sec(\theta))^{-1/2} \).
Take the natural logarithm of both sides to apply logarithmic differentiation: \( \ln y = \ln \theta + \ln \sin(\theta) + \ln (\sec(\theta))^{-1/2} \).
Simplify the logarithm of the power term using the property \( \ln(a^b) = b \ln a \): \( \ln y = \ln \theta + \ln \sin(\theta) - \frac{1}{2} \ln \sec(\theta) \).
Differentiate both sides with respect to \( \theta \), remembering that \( \frac{d}{d\theta} \ln y = \frac{1}{y} \frac{dy}{d\theta} \) by the chain rule. So, \( \frac{1}{y} \frac{dy}{d\theta} = \frac{1}{\theta} + \cot(\theta) - \frac{1}{2} \cdot \frac{d}{d\theta} \ln \sec(\theta) \).
Find the derivative of \( \ln \sec(\theta) \) using the chain rule: \( \frac{d}{d\theta} \ln \sec(\theta) = \tan(\theta) \). Substitute this back into the equation and then solve for \( \frac{dy}{d\theta} \) by multiplying both sides by \( y \).

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Key Concepts

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

Logarithmic Differentiation

Logarithmic differentiation is a technique used to differentiate functions that are products, quotients, or powers of variable expressions. By taking the natural logarithm of both sides, the differentiation process simplifies using properties of logarithms, making it easier to handle complex expressions.
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Logarithmic Differentiation

Properties of Logarithms

The properties of logarithms, such as log(ab) = log a + log b, log(a/b) = log a - log b, and log(a^n) = n log a, allow us to rewrite complicated products, quotients, and powers into sums and differences. These properties are essential for simplifying the function before differentiating.
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Change of Base Property

Derivative of Trigonometric Functions

Understanding the derivatives of trigonometric functions like sin(θ) and sec(θ) is crucial. For example, d/dθ[sin(θ)] = cos(θ) and d/dθ[sec(θ)] = sec(θ)tan(θ). These derivatives are used after applying logarithmic differentiation to find the final derivative of the given function.
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Derivatives of Other Inverse Trigonometric Functions
Related Practice
Textbook Question

Evaluate the integrals in Exercises 111–114.

111. ∫₁^(ln x) (1 / t) dt,x > 1

Textbook Question

Solve the initial value problems in Exercises 55–58.


57. d²y/dx² = 2e^(−x),y(0) = 1,y′(0) = 0

Textbook Question

Evaluate the integrals in Exercises 77–90.

77. ∫dx/√(-x²+4x-3)

Textbook Question

In Exercises 21–48, find the derivative of y with respect to the appropriate variable.

47. y=(arccot(x³))³

Textbook Question

In Exercises 21–48, find the derivative of y with respect to the appropriate variable.

25. y=arcsec(2s+1)

Textbook Question

In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.

y = e^(θ)(sinθ + cosθ)