Find the derivative of the function using the definition of derivative. 1 - 4t G(t) 3 + t G'(t) = State the domain of the function. (Enter your answer using interval notation.) State the domain of its derivative. (Enter your answer using interval notation.)
Find the derivative of the function using the definition of derivative. 1 - 4t G(t) 3 + t G'(t) = State the domain of the function. (Enter your answer using interval notation.) State the domain of its derivative. (Enter your answer using interval notation.)
Chapter3: Functions
Section3.5: Transformation Of Functions
Problem 5SE: How can you determine whether a function is odd or even from the formula of the function?
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![### Calculus Practice Problems: Derivatives
#### Problem Statement
1. **Find the derivative of the function using the definition of derivative.**
Given:
\[
G(t) = \frac{1 - 4t}{3 + t}
\]
Enter your answer in the box provided:
\[
G'(t) = \_\_\_\_\_\_
\]
2. **State the domain of the function.** (Enter your answer using interval notation.)
Enter your answer in the box provided:
\[
\_\_\_\_\_\_
\]
3. **State the domain of its derivative.** (Enter your answer using interval notation.)
Enter your answer in the box provided:
\[
\_\_\_\_\_\_
\]
#### Detailed Explanation
1. **Derivative Concept:**
To find the derivative of the given function \(G(t)\) using the definition of a derivative, apply the limit:
\[
G'(t) = \lim_{h \to 0} \frac{G(t + h) - G(t)}{h}
\]
2. **Domain of the Function:**
The domain of \(G(t)\) represents all the permissible values of \(t\). Examine the denominator \(3 + t\) and ensure it doesn't equal zero (since division by zero is undefined).
3. **Domain of the Derivative:**
After computing the derivative, determine the set of values for which the derivative is defined and valid.
### Required Steps to Solve these Problems:
1. **Calculate the derivative \(G'(t)\) manually or using algebraic simplifications.**
2. **Identify the domain of the function \(G(t)\) considering restrictions from the denominator.**
3. **Determine the domain of \(G'(t)\) using similar principles ensuring the derivative function is defined.**
This structured practice aims to solidify your understanding of derivatives and domains in calculus, applying both definition-based and interval notation concepts.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdda64aa3-7b39-4822-b4bf-c5cb09540d03%2F0264fcfe-6ff1-42f7-a3b8-d58bba3e772e%2Fb37s4f_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculus Practice Problems: Derivatives
#### Problem Statement
1. **Find the derivative of the function using the definition of derivative.**
Given:
\[
G(t) = \frac{1 - 4t}{3 + t}
\]
Enter your answer in the box provided:
\[
G'(t) = \_\_\_\_\_\_
\]
2. **State the domain of the function.** (Enter your answer using interval notation.)
Enter your answer in the box provided:
\[
\_\_\_\_\_\_
\]
3. **State the domain of its derivative.** (Enter your answer using interval notation.)
Enter your answer in the box provided:
\[
\_\_\_\_\_\_
\]
#### Detailed Explanation
1. **Derivative Concept:**
To find the derivative of the given function \(G(t)\) using the definition of a derivative, apply the limit:
\[
G'(t) = \lim_{h \to 0} \frac{G(t + h) - G(t)}{h}
\]
2. **Domain of the Function:**
The domain of \(G(t)\) represents all the permissible values of \(t\). Examine the denominator \(3 + t\) and ensure it doesn't equal zero (since division by zero is undefined).
3. **Domain of the Derivative:**
After computing the derivative, determine the set of values for which the derivative is defined and valid.
### Required Steps to Solve these Problems:
1. **Calculate the derivative \(G'(t)\) manually or using algebraic simplifications.**
2. **Identify the domain of the function \(G(t)\) considering restrictions from the denominator.**
3. **Determine the domain of \(G'(t)\) using similar principles ensuring the derivative function is defined.**
This structured practice aims to solidify your understanding of derivatives and domains in calculus, applying both definition-based and interval notation concepts.
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