Take the logarithm of both sides of the equation and expand it using the product rule of logarithms y=(t)(t+7)(t + 8) In y = In ((t)(t + 7)(t+8)) -0 Now find the derivative of y with respect to t dy dt
Take the logarithm of both sides of the equation and expand it using the product rule of logarithms y=(t)(t+7)(t + 8) In y = In ((t)(t + 7)(t+8)) -0 Now find the derivative of y with respect to t dy dt
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
Question
![**Problem Statement**
Take the logarithm of both sides of the equation and expand it using the product rule of logarithms.
Given:
\[
y = (t)(t + 7)(t + 8)
\]
Taking the natural logarithm of both sides:
\[
\ln y = \ln ((t)(t + 7)(t + 8))
\]
Expand using the product rule:
\[
\ln y = \ln(t) + \ln(t + 7) + \ln(t + 8)
\]
**Differentiation**
Now, find the derivative of \( y \) with respect to \( t \).
\[
\frac{dy}{dt} =
\]
**Explanation**
1. **Logarithmic Expansion:**
- The product rule for logarithms states that the logarithm of a product is the sum of the logarithms of the factors. Hence, \(\ln(abc) = \ln a + \ln b + \ln c\).
2. **Derivative Calculation:**
- To find \(\frac{dy}{dt}\), use implicit differentiation of the logarithmic form or differentiate the product directly.
**Note:** Fill in the blanks with appropriate calculations to complete the solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc8db0c58-8b17-40ce-ad80-341585b72407%2F87b3f632-11ff-4ee9-a38d-815d7c155e81%2Fstsbm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement**
Take the logarithm of both sides of the equation and expand it using the product rule of logarithms.
Given:
\[
y = (t)(t + 7)(t + 8)
\]
Taking the natural logarithm of both sides:
\[
\ln y = \ln ((t)(t + 7)(t + 8))
\]
Expand using the product rule:
\[
\ln y = \ln(t) + \ln(t + 7) + \ln(t + 8)
\]
**Differentiation**
Now, find the derivative of \( y \) with respect to \( t \).
\[
\frac{dy}{dt} =
\]
**Explanation**
1. **Logarithmic Expansion:**
- The product rule for logarithms states that the logarithm of a product is the sum of the logarithms of the factors. Hence, \(\ln(abc) = \ln a + \ln b + \ln c\).
2. **Derivative Calculation:**
- To find \(\frac{dy}{dt}\), use implicit differentiation of the logarithmic form or differentiate the product directly.
**Note:** Fill in the blanks with appropriate calculations to complete the solution.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
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