Textbook Question
10–19. Derivatives Find the derivatives of the following functions.
f(x) = ln(3 sin² 4x)
Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
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Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.R.12
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.R.12Chapter 7, Problem 7.R.12
10–19. Derivatives Find the derivatives of the following functions.
f(x) = (sinh x) / (1 + sinh x)
Verified step by step guidance1
Identify the function to differentiate: \(f(x) = \frac{\sinh x}{1 + \sinh x}\), which is a quotient of two functions.
Recall the Quotient Rule for derivatives: if \(f(x) = \frac{u(x)}{v(x)}\), then \(f'(x) = \frac{u'(x)v(x) - u(x)v'(x)}{(v(x))^2}\).
Set \(u(x) = \sinh x\) and \(v(x) = 1 + \sinh x\). Compute their derivatives: \(u'(x) = \cosh x\) and \(v'(x) = \cosh x\).
Apply the Quotient Rule: substitute \(u\), \(v\), \(u'\), and \(v'\) into the formula to get \(f'(x) = \frac{\cosh x (1 + \sinh x) - \sinh x \cosh x}{(1 + \sinh x)^2}\).
Simplify the numerator by factoring and combining like terms to express the derivative in its simplest form.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Hyperbolic Functions
Hyperbolic functions like sinh(x) are analogs of trigonometric functions but based on hyperbolas. The sinh function is defined as (e^x - e^(-x))/2 and has properties similar to sine, including specific derivatives that are essential for differentiation.
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Asymptotes of Hyperbolas
Quotient Rule
The quotient rule is used to differentiate functions expressed as one function divided by another. It states that the derivative of f(x)/g(x) is (f'(x)g(x) - f(x)g'(x)) / [g(x)]^2, which is crucial for finding the derivative of the given function.
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06:43The Quotient Rule
Derivative of Hyperbolic Sine
The derivative of sinh(x) is cosh(x), another hyperbolic function defined as (e^x + e^(-x))/2. Knowing this derivative is necessary to apply the quotient rule correctly when differentiating the given function.
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03:53Derivatives of Sine & Cosine
Related Practice
Textbook Question
63–68. Definite integrals Evaluate the following definite integrals. Use Theorem 7.7 to express your answer in terms of logarithms.
∫₁/₈¹ dx/x√(1 + x²/³)
Textbook Question
Log-normal probability distribution A commonly used distribution in probability and statistics is the log-normal distribution. (If the logarithm of a variable has a normal distribution, then the variable itself has a log-normal distribution.) The distribution function is
f(x) = 1/xσ√(2π) e⁻ˡⁿ^² ˣ / ²σ^², for x ≥ 0
where ln x has zero mean and standard deviation σ > 0.
b. Evaluate lim x → 0 ƒ(x). (Hint: Let x = eʸ.)
Textbook Question
27–28. Curve sketching Use the graphing techniques of Section 4.4 to graph the following functions on their domains. Identify local extreme points, inflection points, concavity, and end behavior. Use a graphing utility only to check your work.
f(x) = ln x – ln² x
Textbook Question
37–56. Integrals Evaluate each integral.
∫ sech² w tanh w dw
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Textbook Question
10–19. Derivatives Find the derivatives of the following functions.
f(x) = tanh⁻¹(cos x)