Suppose f(x) and f'(x) have the values shown. X 0 1 2 3 f(x) 10 8 9 7 f'(x) 7 8 6 4 5 4 3 لا Let y³ + [f(x)]³ = 10xy + 123. (DO NOT assume y = f(x).) dy dx > Next Question Use implicit differentiation to determine y³ + [ƒ(x)]³ = 10xy + 123. Simplify. at the point (4,-4) on the curve
Suppose f(x) and f'(x) have the values shown. X 0 1 2 3 f(x) 10 8 9 7 f'(x) 7 8 6 4 5 4 3 لا Let y³ + [f(x)]³ = 10xy + 123. (DO NOT assume y = f(x).) dy dx > Next Question Use implicit differentiation to determine y³ + [ƒ(x)]³ = 10xy + 123. Simplify. at the point (4,-4) on the curve
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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![### Implicit Differentiation Practice
#### Problem Statement:
Suppose \( f(x) \) and \( f'(x) \) have the values shown in the table below:
| \( x \) | 0 | 1 | 2 | 3 | 4 |
|----------|---|---|---|---|---|
| \( f(x) \) | 10 | 8 | 9 | 7 | 3 |
| \( f'(x) \) | 7 | 8 | 6 | 4 | 5 |
Given:
\[ y^3 + [f(x)]^3 = 10xy + 123 \text{. (DO NOT assume } y = f(x) \text{).}\]
#### Task:
Use implicit differentiation to determine \(\frac{dy}{dx}\) at the point \((4, -4)\) on the curve:
\[ y^3 + [f(x)]^3 = 10xy + 123. \]
Simplify your result.
#### Approach to Solve:
1. **Implicit Differentiation:**
Apply implicit differentiation to the equation \( y^3 + [f(x)]^3 = 10xy + 123 \) with respect to \( x \).
2. **Find \( \frac{dy}{dx} \) at \( (4, -4) \):**
Substitute \( x = 4 \) and \( y = -4 \) into the differentiated equation and solve for \( \frac{dy}{dx} \).
#### Steps:
1. **Differentiation of \(y^3 + [f(x)]^3\):**
\[ \frac{d}{dx} (y^3) + \frac{d}{dx} ([f(x)]^3) \]
\[ 3y^2 \frac{dy}{dx} + 3[f(x)]^2 f'(x) \]
2. **Differentiation of \(10xy + 123\):**
\[ \frac{d}{dx} (10xy) + \frac{d}{dx} (123) \]
\[ 10y + 10x \frac{dy}{dx} \]
3. **Combine and Simplify:**
\[ 3y^2 \frac{dy}{dx} + 3[f(x)]^2 f'(x) =](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffdc195bb-9009-47db-866e-65b6514806e8%2F0087c1ea-0ea3-4ad3-a5b6-3266383fca75%2F2vcntf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Implicit Differentiation Practice
#### Problem Statement:
Suppose \( f(x) \) and \( f'(x) \) have the values shown in the table below:
| \( x \) | 0 | 1 | 2 | 3 | 4 |
|----------|---|---|---|---|---|
| \( f(x) \) | 10 | 8 | 9 | 7 | 3 |
| \( f'(x) \) | 7 | 8 | 6 | 4 | 5 |
Given:
\[ y^3 + [f(x)]^3 = 10xy + 123 \text{. (DO NOT assume } y = f(x) \text{).}\]
#### Task:
Use implicit differentiation to determine \(\frac{dy}{dx}\) at the point \((4, -4)\) on the curve:
\[ y^3 + [f(x)]^3 = 10xy + 123. \]
Simplify your result.
#### Approach to Solve:
1. **Implicit Differentiation:**
Apply implicit differentiation to the equation \( y^3 + [f(x)]^3 = 10xy + 123 \) with respect to \( x \).
2. **Find \( \frac{dy}{dx} \) at \( (4, -4) \):**
Substitute \( x = 4 \) and \( y = -4 \) into the differentiated equation and solve for \( \frac{dy}{dx} \).
#### Steps:
1. **Differentiation of \(y^3 + [f(x)]^3\):**
\[ \frac{d}{dx} (y^3) + \frac{d}{dx} ([f(x)]^3) \]
\[ 3y^2 \frac{dy}{dx} + 3[f(x)]^2 f'(x) \]
2. **Differentiation of \(10xy + 123\):**
\[ \frac{d}{dx} (10xy) + \frac{d}{dx} (123) \]
\[ 10y + 10x \frac{dy}{dx} \]
3. **Combine and Simplify:**
\[ 3y^2 \frac{dy}{dx} + 3[f(x)]^2 f'(x) =
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