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...
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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
Find the derivative
![The given function is:
\[ g(x) = \ln \left( \frac{(x^2 + 4)(x - 2)^2}{\sqrt{x + 7}} \right) \]
**Task 1: Find the derivative of this function, \( g'(x) \).**
Use the properties of logarithms and differentiation techniques to find the first derivative. Consider using the chain rule and the properties of logarithms such as:
- \(\ln(a/b) = \ln a - \ln b\)
- \(\ln(a^b) = b \ln a\)
**Task 2: Find the second derivative of this function, \( g''(x) \).**
After finding the first derivative \( g'(x) \), apply the rules of differentiation again to find the second derivative. Keep in mind the usage of the product rule and the chain rule as necessary.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F97349252-9d0c-4bc1-b00c-347e0f2af493%2Ff5958708-848f-43d7-a8df-792f9bc5ad17%2Fdsegbg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The given function is:
\[ g(x) = \ln \left( \frac{(x^2 + 4)(x - 2)^2}{\sqrt{x + 7}} \right) \]
**Task 1: Find the derivative of this function, \( g'(x) \).**
Use the properties of logarithms and differentiation techniques to find the first derivative. Consider using the chain rule and the properties of logarithms such as:
- \(\ln(a/b) = \ln a - \ln b\)
- \(\ln(a^b) = b \ln a\)
**Task 2: Find the second derivative of this function, \( g''(x) \).**
After finding the first derivative \( g'(x) \), apply the rules of differentiation again to find the second derivative. Keep in mind the usage of the product rule and the chain rule as necessary.
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