Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
Related questions
Concept explainers
Rate of Change
The relation between two quantities which displays how much greater one quantity is than another is called ratio.
Slope
The change in the vertical distances is known as the rise and the change in the horizontal distances is known as the run. So, the rise divided by run is nothing but a slope value. It is calculated with simple algebraic equations as:
Question
![**Logarithmic Differentiation Example**
**Problem:** Use logarithmic differentiation to find \( y' \) if \( y = (\cos x)^{\cos x} \).
**Solution:**
To differentiate the function \( y = (\cos x)^{\cos x} \), we will use the method of logarithmic differentiation. This technique is especially useful when dealing with functions that are powers of other functions.
1. **Take the natural logarithm of both sides:**
\[
\ln y = \ln \left( (\cos x)^{\cos x} \right)
\]
2. **Simplify using logarithm properties:**
The property \(\ln(a^b) = b \ln a\) gives us:
\[
\ln y = \cos x \cdot \ln (\cos x)
\]
3. **Differentiate both sides with respect to \( x \):**
The left side becomes \(\frac{1}{y} \cdot \frac{dy}{dx}\). For the right side, apply the product rule to \(\cos x \cdot \ln (\cos x)\):
\[
\frac{1}{y} \cdot \frac{dy}{dx} = \left(-\sin x\right) \ln (\cos x) + \frac{-\sin x}{\cos x} \cos x
\]
Simplify the equation:
\[
\frac{1}{y} \cdot \frac{dy}{dx} = -\sin x \cdot \ln (\cos x) - \sin x
\]
4. **Solve for \( \frac{dy}{dx} \):**
Multiply both sides by \( y \) (which is \( (\cos x)^{\cos x} \)):
\[
\frac{dy}{dx} = (\cos x)^{\cos x} \left( -\sin x \cdot \ln (\cos x) - \sin x \right)
\]
Therefore, the derivative \( y' \) is:
\[
y' = - (\cos x)^{\cos x} \cdot \sin x \cdot \left( \ln (\cos x) + 1 \right)
\]
This solution demonstrates the application of logarithmic differentiation to find](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F15976126-8979-495c-a2c3-d70684fdeb4f%2Fd67d68c4-7b37-4ffb-bdb0-626c0f07a80e%2Fdl36fx9_processed.png&w=3840&q=75)
Transcribed Image Text:**Logarithmic Differentiation Example**
**Problem:** Use logarithmic differentiation to find \( y' \) if \( y = (\cos x)^{\cos x} \).
**Solution:**
To differentiate the function \( y = (\cos x)^{\cos x} \), we will use the method of logarithmic differentiation. This technique is especially useful when dealing with functions that are powers of other functions.
1. **Take the natural logarithm of both sides:**
\[
\ln y = \ln \left( (\cos x)^{\cos x} \right)
\]
2. **Simplify using logarithm properties:**
The property \(\ln(a^b) = b \ln a\) gives us:
\[
\ln y = \cos x \cdot \ln (\cos x)
\]
3. **Differentiate both sides with respect to \( x \):**
The left side becomes \(\frac{1}{y} \cdot \frac{dy}{dx}\). For the right side, apply the product rule to \(\cos x \cdot \ln (\cos x)\):
\[
\frac{1}{y} \cdot \frac{dy}{dx} = \left(-\sin x\right) \ln (\cos x) + \frac{-\sin x}{\cos x} \cos x
\]
Simplify the equation:
\[
\frac{1}{y} \cdot \frac{dy}{dx} = -\sin x \cdot \ln (\cos x) - \sin x
\]
4. **Solve for \( \frac{dy}{dx} \):**
Multiply both sides by \( y \) (which is \( (\cos x)^{\cos x} \)):
\[
\frac{dy}{dx} = (\cos x)^{\cos x} \left( -\sin x \cdot \ln (\cos x) - \sin x \right)
\]
Therefore, the derivative \( y' \) is:
\[
y' = - (\cos x)^{\cos x} \cdot \sin x \cdot \left( \ln (\cos x) + 1 \right)
\]
This solution demonstrates the application of logarithmic differentiation to find
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.Recommended textbooks for you
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781285741550
Author:
James Stewart
Publisher:
Cengage Learning
Thomas' Calculus (14th Edition)
Calculus
ISBN:
9780134438986
Author:
Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:
PEARSON
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:
9780134763644
Author:
William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:
PEARSON
Calculus: Early Transcendentals
Calculus
ISBN:
9781319050740
Author:
Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:
W. H. Freeman
Calculus: Early Transcendental Functions
Calculus
ISBN:
9781337552516
Author:
Ron Larson, Bruce H. Edwards
Publisher:
Cengage Learning