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
I need help calculating the derivitve
![The problem asks us to find \( y' \) if \( y = \log_2 \left( \frac{8}{\sqrt{4x+1}} \right) \).
To find the derivative \( y' \), you can use the chain rule and properties of logarithms. Let's step through it:
### Step 1: Simplify the expression
Using properties of logarithms, the given expression can be rewritten as:
\[ y = \log_2(8) - \log_2(\sqrt{4x+1}) \]
\[ y = 3 - \frac{1}{2} \log_2(4x+1) \]
### Step 2: Differentiate
We differentiate using the chain rule:
\[ y' = 0 - \frac{1}{2} \cdot \frac{1}{(4x+1) \ln(2)} \cdot 4 \]
### Conclusion
\[ y' = -\frac{2}{(4x+1) \ln(2)} \]
This expression represents the derivative of the original function. Use this derivative to analyze the behavior and rate of change of the function for various values of \( x \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc3d9838a-dc04-415e-b72d-73cac5bd3472%2F7ed42cca-50f7-4377-83fb-0f5018581a2d%2Fxar6tu1_processed.png&w=3840&q=75)
Transcribed Image Text:The problem asks us to find \( y' \) if \( y = \log_2 \left( \frac{8}{\sqrt{4x+1}} \right) \).
To find the derivative \( y' \), you can use the chain rule and properties of logarithms. Let's step through it:
### Step 1: Simplify the expression
Using properties of logarithms, the given expression can be rewritten as:
\[ y = \log_2(8) - \log_2(\sqrt{4x+1}) \]
\[ y = 3 - \frac{1}{2} \log_2(4x+1) \]
### Step 2: Differentiate
We differentiate using the chain rule:
\[ y' = 0 - \frac{1}{2} \cdot \frac{1}{(4x+1) \ln(2)} \cdot 4 \]
### Conclusion
\[ y' = -\frac{2}{(4x+1) \ln(2)} \]
This expression represents the derivative of the original function. Use this derivative to analyze the behavior and rate of change of the function for various values of \( x \).
Expert Solution
Step 1
Given the function .
The objective is to find the derivative .
Formula used:
(1) Chain rule of differentiation is , where and are differentiable function.
(2) .
(3) Quotient rule of differentiation is . where and are differentiable function
(4) Power rule of differentiation is , for all real .
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