Textbook Question
Solve the initial value problems in Exercises 55–58.
57. d²y/dx² = 2e^(−x),y(0) = 1,y′(0) = 0
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 Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.3.15
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = e^(θ)(sinθ + cosθ)
Verified step by step guidance1
Identify the function to differentiate: \(y = e^{\theta}(\sin\theta + \cos\theta)\), which is a product of two functions of \(\theta\).
Recall the product rule for differentiation: if \(y = u(\theta) \cdot v(\theta)\), then \(\frac{dy}{d\theta} = u'(\theta) v(\theta) + u(\theta) v'(\theta)\).
Set \(u(\theta) = e^{\theta}\) and \(v(\theta) = \sin\theta + \cos\theta\). Compute their derivatives separately: \(u'(\theta) = e^{\theta}\) and \(v'(\theta) = \cos\theta - \sin\theta\).
Apply the product rule: \(\frac{dy}{d\theta} = e^{\theta}(\sin\theta + \cos\theta) + e^{\theta}(\cos\theta - \sin\theta)\).
Simplify the expression by combining like terms inside the parentheses to write 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.
Derivative of Exponential Functions
The derivative of an exponential function like e^θ with respect to θ is e^θ itself. This property is fundamental when differentiating expressions where the exponential function is multiplied by other functions.
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Derivatives of General Exponential Functions
Product Rule
The product rule is used to differentiate the product of two functions. It states that the derivative of f(θ)g(θ) is f'(θ)g(θ) + f(θ)g'(θ). This rule is essential when differentiating y = e^θ(sinθ + cosθ).
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05:18The Product Rule
Derivatives of Trigonometric Functions
The derivatives of sinθ and cosθ with respect to θ are cosθ and -sinθ, respectively. Knowing these derivatives is crucial for differentiating the trigonometric part of the function y = e^θ(sinθ + cosθ).
Recommended video:
06:35Derivatives of Other Inverse Trigonometric Functions
Related Practice
Textbook Question
130. Where does the periodic function f(x) = 2e^(sin(x/2)) take on its extreme values, and what are these values?
Textbook Question
In Exercises 57–70, use logarithmic differentiation to find the derivative of y with respect to the given independent variable.
66. y = θsin(θ)/√(sec(θ))
Textbook Question
Evaluate the integrals in Exercises 77–90.
77. ∫dx/√(-x²+4x-3)
Textbook Question
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
47. y=(arccot(x³))³
Textbook Question
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
25. y=arcsec(2s+1)