Textbook Question
10–19. Derivatives Find the derivatives of the following functions.
g(t) = sinh⁻¹(√t)
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.33b
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.R.33bChapter 7, Problem 7.R.33b
Derivatives of hyperbolic functions Compute the following derivatives.
b. d/dx (x sech x)
Verified step by step guidance1
Identify the function to differentiate: \(f(x) = x \cdot \text{sech}(x)\), which is a product of two functions, \(x\) and \(\text{sech}(x)\).
Recall the product rule for derivatives: if \(f(x) = u(x) v(x)\), then \(f'(x) = u'(x) v(x) + u(x) v'(x)\).
Compute the derivative of the first function: \(u(x) = x\), so \(u'(x) = 1\).
Compute the derivative of the second function: \(v(x) = \text{sech}(x)\). Use the fact that \(\frac{d}{dx} \text{sech}(x) = -\text{sech}(x) \tanh(x)\).
Apply the product rule: \(f'(x) = 1 \cdot \text{sech}(x) + x \cdot (-\text{sech}(x) \tanh(x))\), then simplify the expression as needed.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative of a Product
When differentiating a product of two functions, use the product rule: (fg)' = f'g + fg'. This rule allows you to find the derivative of expressions like x·sech(x) by differentiating each part separately and combining the results.
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The Product Rule
Derivative of the Hyperbolic Secant Function (sech x)
The hyperbolic secant function, sech(x), is defined as 1/cosh(x). Its derivative is -sech(x)·tanh(x), which is essential to apply when differentiating expressions involving sech(x).
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04:56Derivative of the Natural Exponential Function (e^x)
Hyperbolic Functions and Their Properties
Hyperbolic functions like sinh, cosh, and sech have properties similar to trigonometric functions but relate to exponential functions. Understanding their definitions and relationships helps in differentiating and simplifying expressions involving them.
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06:21Properties of Functions
Related Practice
Textbook Question
Linear approximation Find the linear approximation to ƒ(x) = cosh x at a = ln 3 and then use it to approximate the value of cosh 1.
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.
e. For what value of σ > 0 in part (d) does ƒ(x*) have a minimum?
Textbook Question
Caffeine An adult consumes an espresso containing 75 mg of caffeine. If the caffeine has a half-life of 5.5 hours, when will the amount of caffeine in her bloodstream equal 30 mg?
Textbook Question
Limit Evaluate lim x → ∞ (tanh x)ˣ.
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. ln xy = (ln x)(ln y)