Textbook Question
Evaluate the integrals in Exercises 87–96.
95. ∫₂⁴ x^(2x) (1 + ln x) dx
1
views
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
Not the one you use?Change textbookSelect a different textbook
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 7.7.15
In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
15. y = 2√t tanh(√t)
Verified step by step guidance1
Identify the function to differentiate: \(y = 2 \sqrt{t} \tanh(\sqrt{t})\). Notice that \(y\) is a product of two functions of \(t\): \(2 \sqrt{t}\) and \(\tanh(\sqrt{t})\).
Recall the product rule for derivatives: if \(y = u(t) v(t)\), then \(\frac{dy}{dt} = u'(t) v(t) + u(t) v'(t)\). Here, let \(u(t) = 2 \sqrt{t}\) and \(v(t) = \tanh(\sqrt{t})\).
Find \(u'(t)\): Since \(u(t) = 2 t^{1/2}\), use the power rule to get \(u'(t) = 2 \times \frac{1}{2} t^{-1/2} = t^{-1/2}\).
Find \(v'(t)\): Since \(v(t) = \tanh(\sqrt{t})\), apply the chain rule. First, recall that \(\frac{d}{dx} \tanh(x) = \operatorname{sech}^2(x)\). Let \(g(t) = \sqrt{t} = t^{1/2}\), so \(v(t) = \tanh(g(t))\). Then, \(v'(t) = \operatorname{sech}^2(g(t)) \cdot g'(t)\), where \(g'(t) = \frac{1}{2} t^{-1/2}\).
Combine all parts using the product rule: \(\frac{dy}{dt} = u'(t) v(t) + u(t) v'(t) = t^{-1/2} \tanh(\sqrt{t}) + 2 \sqrt{t} \times \operatorname{sech}^2(\sqrt{t}) \times \frac{1}{2} t^{-1/2}\). Simplify the expression as needed.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative of Composite Functions (Chain Rule)
The chain rule is used to differentiate composite functions, where one function is inside another. It states that the derivative of f(g(x)) is f'(g(x)) multiplied by g'(x). This is essential when differentiating expressions like √t or tanh(√t), which involve nested functions.
Recommended video:
03:58
The Chain Rule for 3+ Functions
Derivative of Hyperbolic Functions
Hyperbolic functions such as tanh(x) have specific derivatives; for example, the derivative of tanh(x) is sech²(x). Knowing these derivatives allows you to differentiate terms like tanh(√t) correctly, especially when combined with the chain rule.
Recommended video:
5:50Asymptotes of Hyperbolas
Product Rule for Differentiation
The product rule is used when differentiating the product of two functions. It states that the derivative of u(t)v(t) is u'(t)v(t) + u(t)v'(t). Since y = 2√t * tanh(√t) is a product of two functions of t, applying the product rule is necessary.
Recommended video:
05:18The Product Rule
Related Practice
Textbook Question
Evaluate the integrals in Exercises 87–96.
87. ∫ 5ˣ dx
Textbook Question
80. Volume The region enclosed by the curve y=sech(x), the x-axis, and the lines x=±ln√3 is revolved about the x-axis to generate a solid. Find the volume of the solid.
1
views
Textbook Question
Evaluate the integrals in Exercises 41–60.
49. ∫(sech(√t)tanh(√t)dt)/√t
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.
120. y = x^(sin x)
Textbook Question
13. When is a polynomial f(x) of at most the order of a polynomial g(x) as x→∞? Give reasons for your answer.
1
views