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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Question
![**Differentiate \( f(t) = 5^{-7t + 8} \)**
\[
f'(t) = \_\_\_\_\_\_
\]
To differentiate this function \( f(t) \), we apply the chain rule and the properties of exponential functions. Let's step through this process:
1. Identify the outer and inner functions. Here, the outer function is \( 5^u \) where \( u = -7t + 8 \).
2. Apply the chain rule: if you have a function of the form \( 5^u \), its derivative is \( 5^u \ln(5) \).
3. Differentiate the inner function \( u = -7t + 8 \) with respect to \( t \). The derivative of \( -7t + 8 \) is \( -7 \).
Let’s combine these steps:
- The outer function derivative: \( 5^{-7t+8} \ln(5) \)
- The inner function derivative: \( -7 \)
Thus, by applying the chain rule:
\[ f'(t) = 5^{-7t+8} \ln(5) \cdot (-7) \]
So the complete derivative is:
\[ f'(t) = -7 \cdot 5^{-7t+8} \ln(5) \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F621ac9f8-2dbb-4715-ab7b-42c3ebc90b44%2F2c46eba3-df4b-4b17-b9f5-55df665df801%2Ft34orlt_processed.png&w=3840&q=75)
Transcribed Image Text:**Differentiate \( f(t) = 5^{-7t + 8} \)**
\[
f'(t) = \_\_\_\_\_\_
\]
To differentiate this function \( f(t) \), we apply the chain rule and the properties of exponential functions. Let's step through this process:
1. Identify the outer and inner functions. Here, the outer function is \( 5^u \) where \( u = -7t + 8 \).
2. Apply the chain rule: if you have a function of the form \( 5^u \), its derivative is \( 5^u \ln(5) \).
3. Differentiate the inner function \( u = -7t + 8 \) with respect to \( t \). The derivative of \( -7t + 8 \) is \( -7 \).
Let’s combine these steps:
- The outer function derivative: \( 5^{-7t+8} \ln(5) \)
- The inner function derivative: \( -7 \)
Thus, by applying the chain rule:
\[ f'(t) = 5^{-7t+8} \ln(5) \cdot (-7) \]
So the complete derivative is:
\[ f'(t) = -7 \cdot 5^{-7t+8} \ln(5) \]
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