Textbook Question
37. Plutonium-239 The half-life of the plutonium isotope is 24,360 years. If 10 g of plutonium is released into the atmosphere by a nuclear accident, how many years will it take for 80% of the isotope to decay?
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 7.3.23
In Exercises 7–26, find the derivative of y with respect to x, t, or θ, as appropriate.
y = e^(cost+lnt)
Verified step by step guidance1
Identify the variable with respect to which you need to differentiate. Since the function is given as \(y = e^{\cos t + \ln t}\), and the expression involves \(t\), we will differentiate with respect to \(t\).
Recall the chain rule for differentiation: if \(y = e^{u(t)}\), then \(\frac{dy}{dt} = e^{u(t)} \cdot \frac{du}{dt}\), where \(u(t) = \cos t + \ln t\) in this case.
Find the derivative of the exponent function \(u(t) = \cos t + \ln t\). Differentiate each term separately: \(\frac{d}{dt}(\cos t) = -\sin t\) and \(\frac{d}{dt}(\ln t) = \frac{1}{t}\).
Combine the derivatives of the terms to get \(\frac{du}{dt} = -\sin t + \frac{1}{t}\).
Write the final expression for the derivative using the chain rule: \(\frac{dy}{dt} = e^{\cos t + \ln t} \cdot \left(-\sin t + \frac{1}{t}\right)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Chain Rule
The chain rule is used to differentiate composite functions. It states that the derivative of a function composed of another function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. This is essential when differentiating expressions like e^(cos t + ln t).
Recommended video:
05:02
Intro to the Chain Rule
Derivatives of Exponential Functions
The derivative of an exponential function e^u, where u is a function of the variable, is e^u multiplied by the derivative of u. Recognizing this helps in differentiating y = e^(cos t + ln t) by first identifying the exponent as a function and then applying the chain rule.
Recommended video:
04:50Derivatives of General Exponential Functions
Derivatives of Trigonometric and Logarithmic Functions
Knowing the derivatives of cos t and ln t is crucial. The derivative of cos t with respect to t is -sin t, and the derivative of ln t is 1/t. These derivatives are used when applying the chain rule to find the derivative of the exponent in the given function.
Recommended video:
05:18Derivative of the Natural Logarithmic Function
Related Practice
Textbook Question
Indeterminate Powers and Products
Find the limits in Exercises 53–68.
63. lim (x → ∞) ((x + 2)/(x - 1))^x
Textbook Question
In Exercises 115–126, use logarithmic differentiation or the method in Example 6 to find the derivative of y with respect to the given independent variable.
124. x^(sin y) = ln y
Textbook Question
132. What is special about the functions
f(x) = arcsin((1/√(x²+1)) and g(x)=arctan(1/x)?
Explain.
Textbook Question
Use l’Hôpital’s rule to find the limits in Exercises 7–52.
35. lim (x → 0⁺) ln(x² + 2x) / ln x
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Textbook Question
Suppose that the range of g lies in the domain of f so that the composition fog is defined. If f and g are one-to-one, can anything be said about fog? Give reasons for your answer.