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 Example
**Problem:**
Find \(\frac{dy}{dx}\) by implicit differentiation.
\[ \sqrt{xy} = x^9 y + 54 \]
**Instructions:**
In this example, we will use implicit differentiation to find the derivative \(\frac{dy}{dx}\) of the given equation. Let's proceed step by step.
1. **Rewrite the Given Equation**
\[ \sqrt{xy} = x^9 y + 54 \]
2. **Differentiate Both Sides with Respect to \(x\)**
- For the left side of the equation, we apply the chain rule on \(\sqrt{xy}\).
- For the right side of the equation, apply the product rule to \(x^9 y\) and differentiate the constant \(54\), which gives 0.
3. **Applying Chain Rule to the Left Side**
\[ \frac{d}{dx} (\sqrt{xy}) = \frac{1}{2\sqrt{xy}} \cdot \frac{d}{dx} (xy) \]
4. **Differentiate \(xy\) Using Product Rule**
\[ \frac{d}{dx} (xy) = y + x \frac{dy}{dx} \]
Thus,
\[ \frac{d}{dx} (\sqrt{xy}) = \frac{1}{2\sqrt{xy}} (y + x \frac{dy}{dx}) \]
5. **Applying Product Rule to the Right Side**
\[ \frac{d}{dx} (x^9 y) = 9x^8 y + x^9 \frac{dy}{dx} \]
6. **Differentiate the Constant Term**
\[ \frac{d}{dx} 54 = 0 \]
Putting it all together, we get:
\[ \frac{1}{2\sqrt{xy}} (y + x \frac{dy}{dx}) = 9x^8 y + x^9 \frac{dy}{dx} \]
7. **Simplify and Solve for \(\frac{dy}{dx}\)**
\[ \frac{y + x \frac{dy}{dx}}{2\sqrt{xy}} = 9x^8 y + x^9 \frac{dy}{dx} \]
Multiply both sides by \(2\sqrt{xy}\) to](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc6e1a3f1-b538-4464-8269-7761edea80d0%2F6f9b29f5-e59f-454b-b3a2-e92af891ffb4%2F5onz6ak_processed.png&w=3840&q=75)
Transcribed Image Text:### Implicit Differentiation Example
**Problem:**
Find \(\frac{dy}{dx}\) by implicit differentiation.
\[ \sqrt{xy} = x^9 y + 54 \]
**Instructions:**
In this example, we will use implicit differentiation to find the derivative \(\frac{dy}{dx}\) of the given equation. Let's proceed step by step.
1. **Rewrite the Given Equation**
\[ \sqrt{xy} = x^9 y + 54 \]
2. **Differentiate Both Sides with Respect to \(x\)**
- For the left side of the equation, we apply the chain rule on \(\sqrt{xy}\).
- For the right side of the equation, apply the product rule to \(x^9 y\) and differentiate the constant \(54\), which gives 0.
3. **Applying Chain Rule to the Left Side**
\[ \frac{d}{dx} (\sqrt{xy}) = \frac{1}{2\sqrt{xy}} \cdot \frac{d}{dx} (xy) \]
4. **Differentiate \(xy\) Using Product Rule**
\[ \frac{d}{dx} (xy) = y + x \frac{dy}{dx} \]
Thus,
\[ \frac{d}{dx} (\sqrt{xy}) = \frac{1}{2\sqrt{xy}} (y + x \frac{dy}{dx}) \]
5. **Applying Product Rule to the Right Side**
\[ \frac{d}{dx} (x^9 y) = 9x^8 y + x^9 \frac{dy}{dx} \]
6. **Differentiate the Constant Term**
\[ \frac{d}{dx} 54 = 0 \]
Putting it all together, we get:
\[ \frac{1}{2\sqrt{xy}} (y + x \frac{dy}{dx}) = 9x^8 y + x^9 \frac{dy}{dx} \]
7. **Simplify and Solve for \(\frac{dy}{dx}\)**
\[ \frac{y + x \frac{dy}{dx}}{2\sqrt{xy}} = 9x^8 y + x^9 \frac{dy}{dx} \]
Multiply both sides by \(2\sqrt{xy}\) to
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