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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![**Problem Statement:**
Given that:
\( f(x) = x^7 h(x) \)
\( h(-1) = 5 \)
\( h'(-1) = 8 \)
Calculate \( f'(-1) \).
**Solution:**
To solve this, we use the product rule for derivatives. The product rule states that if \( f(x) = u(x)v(x) \), then the derivative \( f'(x) \) is:
\[ f'(x) = u'(x)v(x) + u(x)v'(x) \]
In this case, let \( u(x) = x^7 \) and \( v(x) = h(x) \).
1. Calculate \( u'(x) \):
\[ u(x) = x^7 \]
\[ u'(x) = 7x^6 \]
2. Evaluate \( u'(-1) \):
\[ u'(-1) = 7(-1)^6 = 7 \]
3. Plug in the values into the product rule:
\[
f'(-1) = u'(-1)v(-1) + u(-1)v'(-1)
\]
Substituting the given values:
\[
f'(-1) = 7 \cdot 5 + (-1)^7 \cdot 8
\]
\[
f'(-1) = 35 - 8
\]
\[
f'(-1) = 27
\]
Thus, \( f'(-1) = 27 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F437d091c-bafd-402f-b1ab-0058f5d0273f%2F8bba0058-b3c7-4697-a76f-2f8cdd2818b8%2Fc0z0b2h_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Given that:
\( f(x) = x^7 h(x) \)
\( h(-1) = 5 \)
\( h'(-1) = 8 \)
Calculate \( f'(-1) \).
**Solution:**
To solve this, we use the product rule for derivatives. The product rule states that if \( f(x) = u(x)v(x) \), then the derivative \( f'(x) \) is:
\[ f'(x) = u'(x)v(x) + u(x)v'(x) \]
In this case, let \( u(x) = x^7 \) and \( v(x) = h(x) \).
1. Calculate \( u'(x) \):
\[ u(x) = x^7 \]
\[ u'(x) = 7x^6 \]
2. Evaluate \( u'(-1) \):
\[ u'(-1) = 7(-1)^6 = 7 \]
3. Plug in the values into the product rule:
\[
f'(-1) = u'(-1)v(-1) + u(-1)v'(-1)
\]
Substituting the given values:
\[
f'(-1) = 7 \cdot 5 + (-1)^7 \cdot 8
\]
\[
f'(-1) = 35 - 8
\]
\[
f'(-1) = 27
\]
Thus, \( f'(-1) = 27 \).
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