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
![**Problem:**
Find \((f^{-1})'(a)\).
Given:
\[ f(x) = 2x^3 + 3x^2 + 4x + 2, \quad a = 2 \]
\[ (f^{-1})'(a) = \boxed{\phantom{answer}} \]
**Explanation:**
We are tasked with finding the derivative of the inverse function \(f^{-1}\) at \(a = 2\). The original function \(f(x)\) is a cubic polynomial. To find \((f^{-1})'(a)\), we will need to use the formula for the derivative of an inverse function:
\[
(f^{-1})'(a) = \frac{1}{f'(f^{-1}(a))}
\]
First, calculate \(f'(x)\), the derivative of \(f(x)\):
\[
f'(x) = \frac{d}{dx}(2x^3 + 3x^2 + 4x + 2) = 6x^2 + 6x + 4
\]
Next steps would typically involve finding \(f^{-1}(a)\), which requires solving for \(x\) such that \(f(x) = a\), and then evaluating the reciprocal of the derivative at that point.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F026225a1-ee17-408a-9754-c8442d45a552%2F55ef6bf7-7cfb-46f5-8bae-02f214035be0%2Frwemun_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem:**
Find \((f^{-1})'(a)\).
Given:
\[ f(x) = 2x^3 + 3x^2 + 4x + 2, \quad a = 2 \]
\[ (f^{-1})'(a) = \boxed{\phantom{answer}} \]
**Explanation:**
We are tasked with finding the derivative of the inverse function \(f^{-1}\) at \(a = 2\). The original function \(f(x)\) is a cubic polynomial. To find \((f^{-1})'(a)\), we will need to use the formula for the derivative of an inverse function:
\[
(f^{-1})'(a) = \frac{1}{f'(f^{-1}(a))}
\]
First, calculate \(f'(x)\), the derivative of \(f(x)\):
\[
f'(x) = \frac{d}{dx}(2x^3 + 3x^2 + 4x + 2) = 6x^2 + 6x + 4
\]
Next steps would typically involve finding \(f^{-1}(a)\), which requires solving for \(x\) such that \(f(x) = a\), and then evaluating the reciprocal of the derivative at that point.
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