Textbook Question
Evaluate the integrals in Exercises 39–56.
55. ∫dx/(2√x + 2x)
25
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 7.3.71
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
71. y = log₂(5θ)
Verified step by step guidance1
Recall the formula for the derivative of a logarithm with an arbitrary base: if \( y = \log_a(u) \), then \( \frac{dy}{dx} = \frac{1}{u \ln(a)} \cdot \frac{du}{dx} \), where \( \ln \) is the natural logarithm.
Identify the base \( a \) and the argument \( u \) in the given function. Here, \( y = \log_2(5\theta) \), so \( a = 2 \) and \( u = 5\theta \).
Compute the derivative of the inside function \( u = 5\theta \) with respect to \( \theta \). Since \( 5 \) is a constant, \( \frac{du}{d\theta} = 5 \).
Apply the derivative formula: \( \frac{dy}{d\theta} = \frac{1}{5\theta \ln(2)} \times 5 \).
Simplify the expression by canceling common factors if possible to express the derivative in its simplest form.
Verified video answer for a similar problem:
This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2mWas this helpful?
Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Derivative of Logarithmic Functions
The derivative of a logarithmic function depends on its base. For a logarithm with base a, log_a(x), the derivative is 1/(x ln(a)) times the derivative of x. Understanding this allows you to differentiate log₂(5θ) by applying the chain rule.
Recommended video:
05:18
Derivative of the Natural Logarithmic Function
Chain Rule
The chain rule is used to differentiate composite functions. When a function is inside another, like 5θ inside the logarithm, you first differentiate the outer function with respect to the inner function, then multiply by the derivative of the inner function.
Recommended video:
05:02Intro to the Chain Rule
Properties of Logarithms
Logarithmic properties, such as log_a(bx) = log_a(b) + log_a(x), can simplify differentiation. Recognizing these properties helps break down complex logarithmic expressions into simpler parts, making differentiation more straightforward.
Recommended video:
05:36Change of Base Property
Related Practice
Textbook Question
Evaluate the integrals in Exercises 33–54.
∫(from ln4 to ln9)e^(x/2)dx
40
views
Textbook Question
41. Cooling soup Suppose that a cup of soup cooled from 90°C to 60°C after 10 min in a room where the temperature was 20°C. Use Newton’s Law of Cooling to answer the following questions.
a. How much longer would it take the soup to cool to 35°C?
5
views
Textbook Question
7. Order the following functions from slowest growing to fastest growing as x→∞.
a. e^x
b. x^x
c. (ln x)^x
d. e^(x/2)
26
views
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.
69. lim (x → ∞) (√(9x + 1)) / (√(x + 1))
30
views
Textbook Question
In Exercises 59–86, find the derivative of y with respect to the given independent variable.
73. y = log₄ x + log₄ x²
26
views