Textbook Question
Evaluate the integrals in Exercises 33–54.
∫(from ln4 to ln9)e^(x/2)dx
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 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
Chapter 7, Problem 7.3.117
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.
117. y = (√t)ᵗ
Verified step by step guidance1
Rewrite the function to a form that is easier to differentiate using logarithmic differentiation. Start by expressing the function as \(y = (\sqrt{t})^{t} = (t^{1/2})^{t} = t^{t/2}\).
Take the natural logarithm of both sides to simplify the exponentiation: \(\ln y = \ln \left(t^{t/2}\right)\).
Use the logarithm power rule to bring down the exponent: \(\ln y = \frac{t}{2} \ln t\).
Differentiate both sides with respect to \(t\). Remember that \(y\) is a function of \(t\), so use implicit differentiation on the left side: \(\frac{1}{y} \frac{dy}{dt} = \frac{d}{dt} \left( \frac{t}{2} \ln t \right)\).
Apply the product rule to differentiate the right side: \(\frac{d}{dt} \left( \frac{t}{2} \ln t \right) = \frac{1}{2} \ln t + \frac{t}{2} \cdot \frac{1}{t}\). Then multiply both sides by \(y\) to solve for \(\frac{dy}{dt}\).
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
4mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Logarithmic Differentiation
Logarithmic differentiation is a technique used to differentiate functions where the variable appears both in the base and the exponent. By taking the natural logarithm of both sides, the expression simplifies, allowing the use of implicit differentiation and product or chain rules more easily.
Recommended video:
06:30
Logarithmic Differentiation
Chain Rule
The chain rule is a fundamental differentiation rule used when differentiating 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.
Recommended video:
05:02Intro to the Chain Rule
Properties of Logarithms
Properties of logarithms, such as log(a^b) = b log(a), help simplify expressions before differentiation. Applying these properties transforms complicated exponentials into products, making differentiation more straightforward.
Recommended video:
05:36Change of Base Property
Related Practice
Textbook Question
Evaluate the integrals in Exercises 33–54.
∫ 2t e^(-t²) dt
Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
83. y = 3^(log₂ t)
Textbook Question
Theory and Applications
L’Hôpital’s Rule does not help with the limits in Exercises 69–76.
Try it—you just keep on cycling. Find the limits some other way.
71. lim (x → (π/2)⁻) sec x / tan x
1
views
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
Indeterminate Powers and Products
Find the limits in Exercises 53–68.
57. lim (x → 0⁺) x^(-1/ln x)