Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter3: The Derivative
Section3.CR: Chapter 3 Review
Problem 8CR
Related questions
Question
![**Problem Statement:**
Find the derivative of the following function.
\[ y = 4 \cdot 5^x \]
**Solution:**
To find the derivative of this function, \( y = 4 \cdot 5^x \), we will use the chain rule and the derivative rule for exponential functions.
The original function can be written as:
\[ y = 4 \cdot 5^x \]
To differentiate \( y = 4 \cdot 5^x \), we use the property of derivatives that states \( \frac{d}{dx} a^x = a^x \ln(a) \) where \( a \) is a constant. Here, \( a = 5 \).
Step-by-Step Differentiation:
1. Identify the constants and the variable function:
\[ y = 4 \cdot 5^x \]
Here, 4 is a constant, and \( 5^x \) is the variable part.
2. Differentiate \( 5^x \) with respect to \( x \):
\[ \frac{d}{dx} 5^x = 5^x \ln(5) \]
3. Apply the constant multiple rule, which allows us to take the constant (4) outside the differentiation operator:
\[ y' = 4 \cdot \frac{d}{dx} (5^x) \]
\[ y' = 4 \cdot 5^x \ln(5) \]
Hence, the derivative of the function \( y = 4 \cdot 5^x \) is:
\[ \boxed{y' = 4 \cdot 5^x \ln(5)} \]
This is the final answer and provides the rate of change of the given function with respect to \( x \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F70018f71-7185-4cb2-be6c-94aac78b8d2f%2F2c21eed9-6dd6-453b-b47e-f42a171956c9%2F856rhcq_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Find the derivative of the following function.
\[ y = 4 \cdot 5^x \]
**Solution:**
To find the derivative of this function, \( y = 4 \cdot 5^x \), we will use the chain rule and the derivative rule for exponential functions.
The original function can be written as:
\[ y = 4 \cdot 5^x \]
To differentiate \( y = 4 \cdot 5^x \), we use the property of derivatives that states \( \frac{d}{dx} a^x = a^x \ln(a) \) where \( a \) is a constant. Here, \( a = 5 \).
Step-by-Step Differentiation:
1. Identify the constants and the variable function:
\[ y = 4 \cdot 5^x \]
Here, 4 is a constant, and \( 5^x \) is the variable part.
2. Differentiate \( 5^x \) with respect to \( x \):
\[ \frac{d}{dx} 5^x = 5^x \ln(5) \]
3. Apply the constant multiple rule, which allows us to take the constant (4) outside the differentiation operator:
\[ y' = 4 \cdot \frac{d}{dx} (5^x) \]
\[ y' = 4 \cdot 5^x \ln(5) \]
Hence, the derivative of the function \( y = 4 \cdot 5^x \) is:
\[ \boxed{y' = 4 \cdot 5^x \ln(5)} \]
This is the final answer and provides the rate of change of the given function with respect to \( x \).
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