Domanda del libro di testo
10–19. Derivatives Find the derivatives of the following functions.
f(x) = ln(3 sin² 4x)
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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
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Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblema 7.R.12
Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblema 7.R.12Capitolo 7, Problema 7.R.12
10–19. Derivatives Find the derivatives of the following functions.
f(x) = (sinh x) / (1 + sinh x)
Guida verificata passo dopo passo1
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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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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Percorso guidato
5:50
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
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