Textbook Question
In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.
3. y = 1/x ∫(from 1 to x) e^t/t dt, x²y' + xy = e^x
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 7.7.21
In Exercises 13–24, find the derivative of y with respect to the appropriate variable.
21. y = ln(cosh v) - 1/2 tanh²v
Verified step by step guidance1
Identify the function to differentiate: \(y = \ln(\cosh v) - \frac{1}{2} \tanh^{2} v\).
Recall the derivative formulas needed: the derivative of \(\ln(u)\) is \(\frac{1}{u} \frac{du}{dv}\), and the derivative of \(\tanh v\) is \(\operatorname{sech}^2 v\).
Differentiate the first term: \(\frac{d}{dv} \ln(\cosh v) = \frac{1}{\cosh v} \cdot \sinh v\) because \(\frac{d}{dv} \cosh v = \sinh v\).
Differentiate the second term using the chain rule: \(\frac{d}{dv} \left( \frac{1}{2} \tanh^{2} v \right) = \frac{1}{2} \cdot 2 \tanh v \cdot \operatorname{sech}^2 v\).
Combine the derivatives from both terms carefully, remembering to subtract the derivative of the second term from the first to find \(\frac{dy}{dv}\).
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 Hyperbolic Functions
Hyperbolic functions such as sinh, cosh, and tanh have derivatives similar to trigonometric functions but with distinct signs. For example, the derivative of cosh(v) is sinh(v), and the derivative of tanh(v) is sech²(v). Understanding these derivatives is essential for differentiating expressions involving hyperbolic functions.
Recommended video:
5:50
Asymptotes of Hyperbolas
Chain Rule
The chain rule is a fundamental differentiation technique used when dealing with composite functions. It states that the derivative of a composite function is the derivative of the outer function evaluated at the inner function times the derivative of the inner function. This rule is crucial when differentiating functions like ln(cosh v) where cosh v is inside the logarithm.
Recommended video:
05:02Intro to the Chain Rule
Derivative of Logarithmic Functions
The derivative of the natural logarithm function ln(u) with respect to its variable is 1/u times the derivative of u. This rule allows us to differentiate expressions like ln(cosh v) by first identifying the inner function and then applying the chain rule to find the overall derivative.
Recommended video:
05:18Derivative of the Natural Logarithmic Function
Related Practice
Textbook Question
In Exercises 21–48, find the derivative of y with respect to the appropriate variable.
35. y=arccsc(e^t)
Textbook Question
Evaluate the integrals in Exercises 33–54.
51. ∫ from ln(π/6) to ln(π/2) 2e^v cos(e^v) dv
Textbook Question
Rewrite the expressions in Exercises 5–10 in terms of exponentials and simplify the results as much as you can.
6. sinh(2ln x)
Textbook Question
82. For what values of a and b is
lim(x→0)(tan(2x/x³) + a/x² + sin(bx)/x) = 0?
1
views
Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
65. y = (cos θ)^(√2)