Domanda del libro di testo
Find the values in Exercises 9–12.
11. tan(arcsin(-1/2))
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Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- 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
Capitolo 7, Problema 7.7.27
In Exercises 25–36, find the derivative of y with respect to the appropriate variable.
27. y = (1 - θ)tanh⁻¹(θ)
Guida verificata passo dopo passo1
Identify the function given: \(y = (1 - \theta) \tanh^{-1}(\theta)\), where \(\tanh^{-1}(\theta)\) is the inverse hyperbolic tangent function of \(\theta\).
Recognize that \(y\) is a product of two functions of \(\theta\): \(u = (1 - \theta)\) and \(v = \tanh^{-1}(\theta)\). To find \(\frac{dy}{d\theta}\), apply the product rule: \(\frac{dy}{d\theta} = u'v + uv'\).
Compute the derivative of the first function: \(u' = \frac{d}{d\theta}(1 - \theta) = -1\).
Compute the derivative of the second function: \(v' = \frac{d}{d\theta} \tanh^{-1}(\theta) = \frac{1}{1 - \theta^2}\), which is the standard derivative formula for the inverse hyperbolic tangent.
Substitute \(u\), \(u'\), \(v\), and \(v'\) into the product rule formula: \(\frac{dy}{d\theta} = (-1) \cdot \tanh^{-1}(\theta) + (1 - \theta) \cdot \frac{1}{1 - \theta^2}\). This expression represents the derivative of \(y\) with respect to \(\theta\).
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Inverse Hyperbolic Tangent Function (tanh⁻¹)
The inverse hyperbolic tangent function, denoted as tanh⁻¹(x), is the inverse of the hyperbolic tangent function. It is defined for values of x between -1 and 1 and has the derivative d/dx[tanh⁻¹(x)] = 1/(1 - x²). Understanding its domain and derivative is essential for differentiating expressions involving tanh⁻¹.
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3:17
Inverse Tangent
Product Rule of Differentiation
The product rule is used to differentiate functions that are products of two or more functions. If y = u(x)v(x), then dy/dx = u'(x)v(x) + u(x)v'(x). Applying this rule correctly is crucial when differentiating y = (1 - θ) * tanh⁻¹(θ), where both factors depend on θ.
Video consigliato:
05:18The Product Rule
Chain Rule and Variable Differentiation
The chain rule helps differentiate composite functions by multiplying the derivative of the outer function by the derivative of the inner function. Additionally, recognizing the variable with respect to which differentiation is performed (here, θ) ensures correct application of derivative formulas and variable treatment.
Video consigliato:
05:02Intro to the Chain Rule
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