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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![### Derivatives of Functions
#### Problem Statement:
Find the derivative of the function:
\[ y = 5x^2 + 2^x + 3 \]
The image contains a section at the top, where the problem statement and the function to differentiate are provided.
#### Step-by-Step Solution:
To find the derivative of the function \( y = 5x^2 + 2^x + 3 \), we need to apply the basic rules of differentiation.
1. **Constant Rule**: The derivative of a constant is 0.
2. **Power Rule**: The derivative of \( x^n \) is \( nx^{n-1} \).
3. **Exponential Rule**: The derivative of \( a^x \) is \( a^x \ln(a) \).
Let's apply these rules to each term in the function.
**Term-by-Term Differentiation:**
1. For \( 5x^2 \):
\[ \frac{d}{dx}(5x^2) = 5 \cdot 2x^{2-1} = 10x \]
2. For \( 2^x \):
\[ \frac{d}{dx}(2^x) = 2^x \ln(2) \]
3. For \( 3 \):
\[ \frac{d}{dx}(3) = 0 \]
Combining all the results, we get the derivative of the function:
\[ \frac{dy}{dx} = 10x + 2^x \ln(2) \]
Thus, the derivative of \( y = 5x^2 + 2^x + 3 \) is:
\[ \frac{dy}{dx} = 10x + 2^x \ln(2) \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F500c0e80-687a-4e1b-8f8e-54233bb713dd%2F4c6d4866-00b7-4d29-8344-bfa4f7ebdda7%2Ffjx7s_processed.png&w=3840&q=75)
Transcribed Image Text:### Derivatives of Functions
#### Problem Statement:
Find the derivative of the function:
\[ y = 5x^2 + 2^x + 3 \]
The image contains a section at the top, where the problem statement and the function to differentiate are provided.
#### Step-by-Step Solution:
To find the derivative of the function \( y = 5x^2 + 2^x + 3 \), we need to apply the basic rules of differentiation.
1. **Constant Rule**: The derivative of a constant is 0.
2. **Power Rule**: The derivative of \( x^n \) is \( nx^{n-1} \).
3. **Exponential Rule**: The derivative of \( a^x \) is \( a^x \ln(a) \).
Let's apply these rules to each term in the function.
**Term-by-Term Differentiation:**
1. For \( 5x^2 \):
\[ \frac{d}{dx}(5x^2) = 5 \cdot 2x^{2-1} = 10x \]
2. For \( 2^x \):
\[ \frac{d}{dx}(2^x) = 2^x \ln(2) \]
3. For \( 3 \):
\[ \frac{d}{dx}(3) = 0 \]
Combining all the results, we get the derivative of the function:
\[ \frac{dy}{dx} = 10x + 2^x \ln(2) \]
Thus, the derivative of \( y = 5x^2 + 2^x + 3 \) is:
\[ \frac{dy}{dx} = 10x + 2^x \ln(2) \]
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