Textbook Question
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
13. y = (x+2)^(x+2)
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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
Chapter 7, Problem 7.P.5
In Exercises 1–24, find the derivative of y with respect to the appropriate variable.
5. y = ln(sin²θ)
Verified step by step guidance1
Recognize that the function is a composition of functions: \(y = \ln(\sin^{2}\theta)\), which can be rewritten as \(y = \ln((\sin\theta)^{2})\).
Use the logarithmic property to simplify the expression: \(\ln(a^{b}) = b \ln(a)\), so rewrite \(y\) as \(y = 2 \ln(\sin\theta)\).
Differentiate \(y\) with respect to \(\theta\) using the chain rule. The derivative of \(\ln(u)\) with respect to \(\theta\) is \(\frac{1}{u} \cdot \frac{du}{d\theta}\), where \(u = \sin\theta\).
Calculate \(\frac{du}{d\theta}\) where \(u = \sin\theta\), so \(\frac{du}{d\theta} = \cos\theta\).
Combine the results to write the derivative: \(\frac{dy}{d\theta} = 2 \cdot \frac{1}{\sin\theta} \cdot \cos\theta\).
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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 method for finding the derivative of a composite function. It states that the derivative of f(g(x)) is f'(g(x)) multiplied by g'(x). This rule is essential when differentiating functions like ln(sin²θ), where one function is nested inside another.
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Intro to the Chain Rule
Derivative of the Natural Logarithm Function
The derivative of ln(u), where u is a differentiable function of x, is (1/u) times the derivative of u. This property allows us to differentiate logarithmic functions by first identifying the inner function and then applying the chain rule.
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05:18Derivative of the Natural Logarithmic Function
Derivative of Trigonometric Functions
Understanding the derivatives of basic trigonometric functions is crucial. For example, the derivative of sin(θ) with respect to θ is cos(θ). When differentiating sin²θ, the chain rule applies again, treating sin²θ as (sin θ)².
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06:35Derivatives of Other Inverse Trigonometric Functions
Related Practice
Textbook Question
112. True, or false? Give reasons for your answers.
c. ln x = o(x+1)
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Textbook Question
118. A particle is traveling upward and to the right along the curve y=ln(x). Its x-coordinate is increasing at the rate (dx/dt)=√x m/sec. At what rate is the y-coordinate changing at the point (e², 2)?
Textbook Question
In Exercises 25–30, use logarithmic differentiation to find the derivative of y with respect to the appropriate variable.
25. y = 2(x² + 1)/√(cos 2x)
Textbook Question
7. What integrals lead to logarithms? Give examples. What are the integrals of tan x, cot x, sec x, and csc x?
Textbook Question
111. True, or false? Give reasons for your answers.
e. arctan x = O(1)