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
how do i find the value of the derivative? pls show steps so i know
![**Calculating the Derivative of a Function**
This exercise focuses on finding the derivative of a given function and evaluating it at a specified point. The function and set of possible solutions are presented as follows:
**Given Function:**
\[ g(x) = x^3 + 5x \]
Determine \( g'(1) \), the value of the derivative at \( x = 1 \).
**Options:**
1. \( g'(x) = 3x^2 + 5; g'(1) = 8 \)
2. \( g'(x) = x^2 + 5; g'(1) = 6 \)
3. \( g'(x) = 3x^2 + 5x; g'(1) = 8 \)
4. \( g'(x) = 3x^2; g'(1) = 3 \)
### Step-by-Step Solution:
1. **Find the derivative \( g'(x) \):**
\[
g(x) = x^3 + 5x
\]
Using the power rule \( (x^n)' = nx^{n-1} \):
\[
g'(x) = 3x^2 + 5
\]
2. **Evaluate the derivative at \( x = 1 \):**
\[
g'(1) = 3(1)^2 + 5 = 3 \cdot 1 + 5 = 8
\]
Therefore, the correct answer is:
\[ \text{Option 1: } g'(x) = 3x^2 + 5; g'(1) = 8 \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa1de489e-8c51-4ba0-8d15-cf142b4b6c4d%2Fa75053fb-f3b0-4079-9956-e0994fe7bad7%2Faiu55z.jpeg&w=3840&q=75)
Transcribed Image Text:**Calculating the Derivative of a Function**
This exercise focuses on finding the derivative of a given function and evaluating it at a specified point. The function and set of possible solutions are presented as follows:
**Given Function:**
\[ g(x) = x^3 + 5x \]
Determine \( g'(1) \), the value of the derivative at \( x = 1 \).
**Options:**
1. \( g'(x) = 3x^2 + 5; g'(1) = 8 \)
2. \( g'(x) = x^2 + 5; g'(1) = 6 \)
3. \( g'(x) = 3x^2 + 5x; g'(1) = 8 \)
4. \( g'(x) = 3x^2; g'(1) = 3 \)
### Step-by-Step Solution:
1. **Find the derivative \( g'(x) \):**
\[
g(x) = x^3 + 5x
\]
Using the power rule \( (x^n)' = nx^{n-1} \):
\[
g'(x) = 3x^2 + 5
\]
2. **Evaluate the derivative at \( x = 1 \):**
\[
g'(1) = 3(1)^2 + 5 = 3 \cdot 1 + 5 = 8
\]
Therefore, the correct answer is:
\[ \text{Option 1: } g'(x) = 3x^2 + 5; g'(1) = 8 \]
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