Öz Given z = sin(y) where a=t+u and y=t-u, find Ot What is the formula for ✓ Find Oz Ət dz Ot : 8z 8t 8z Ot 8z 8t 8z Ot || || = 8z 8x 8 8t 8z Bu 8t 8t 8z Oy By Ot 8z Du 8z 8u + Bu St 8t 8x + + A Congratulations! your answer is correct (and saved). Check Answer/Save Step-By-Step Example 8z By By Ot 8t By By Ot 8x By By Da + and Live H
Öz Given z = sin(y) where a=t+u and y=t-u, find Ot What is the formula for ✓ Find Oz Ət dz Ot : 8z 8t 8z Ot 8z 8t 8z Ot || || = 8z 8x 8 8t 8z Bu 8t 8t 8z Oy By Ot 8z Du 8z 8u + Bu St 8t 8x + + A Congratulations! your answer is correct (and saved). Check Answer/Save Step-By-Step Example 8z By By Ot 8t By By Ot 8x By By Da + and Live H
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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Transcribed Image Text:Given \( z = \sin(xy) \) where \( x = t + u \) and \( y = t - u \), find \(\frac{\partial z}{\partial t}\) and \(\frac{\partial z}{\partial u}\).
**What is the formula for** \(\frac{\partial z}{\partial t}\):
Options:
1. \(\frac{\partial z}{\partial x} = \frac{\partial z}{\partial x} \cdot \frac{\partial x}{\partial t} + \frac{\partial z}{\partial y} \cdot \frac{\partial y}{\partial t}\)
2. \(\frac{\partial z}{\partial x} = \frac{\partial z}{\partial x} + \frac{\partial y}{\partial t}\)
3. \(\frac{\partial z}{\partial x} = \frac{\partial z}{\partial y} \cdot \frac{\partial y}{\partial t} + \frac{\partial z}{\partial x} \cdot \frac{\partial x}{\partial t}\)
4. \(\frac{\partial z}{\partial x} = \frac{\partial z}{\partial y} \cdot \frac{\partial x}{\partial t} + \frac{\partial z}{\partial x} \cdot \frac{\partial y}{\partial t}\)
The correct answer, as indicated by the checkmark, is:
\(\frac{\partial z}{\partial x} = \frac{\partial z}{\partial x} \cdot \frac{\partial x}{\partial t} + \frac{\partial z}{\partial y} \cdot \frac{\partial y}{\partial t}\)
On the left side, there is a sidebar with three buttons:
- **Check Answer/Save** (green button)
- **Step By Step Example**
- **Live Help**
Congratulations message: "Congratulations! your answer is correct and saved!"
The format presents interactions typical for an educational assessment platform where users match given functions to derivative rules, particularly using the chain rule for partial derivatives.
![**Understanding the Formula for Partial Derivatives**
In this exercise, you are asked to identify the formula for the partial derivative of \( \frac{\partial z}{\partial u} \).
**Options Provided:**
1. \( \frac{\partial z}{\partial x} = \frac{\partial z}{\partial x} \cdot \frac{\partial x}{\partial y} + \frac{\partial z}{\partial y} \cdot \frac{\partial y}{\partial u} \)
2. \( \frac{\partial z}{\partial x} = \frac{\partial z}{\partial x} \cdot \frac{\partial x}{\partial u} + \frac{\partial z}{\partial y} \cdot \frac{\partial y}{\partial u} \)
3. \( \frac{\partial z}{\partial u} = \frac{\partial z}{\partial x} \cdot \frac{\partial x}{\partial u} + \frac{\partial z}{\partial y} \cdot \frac{\partial y}{\partial u} \) (correct answer)
4. \( \frac{\partial z}{\partial x} = \frac{\partial z}{\partial x} \cdot \frac{\partial x}{\partial u} \)
5. \( \frac{\partial z}{\partial y} = \frac{\partial z}{\partial y} \cdot \frac{\partial y}{\partial u} + \frac{\partial z}{\partial u} \cdot \frac{\partial u}{\partial x} \)
**Correct Solution:**
The correct answer is option 3:
\[ \frac{\partial z}{\partial u} = \frac{\partial z}{\partial x} \cdot \frac{\partial x}{\partial u} + \frac{\partial z}{\partial y} \cdot \frac{\partial y}{\partial u} \]
This represents the use of the chain rule in finding the partial derivative of \( z \) with respect to \( u \) by considering how \( z \) depends on intermediate variables \( x \) and \( y \).
**Feedback:**
Once you select the correct formula, you will receive a congratulatory message: "Congratulations, your answer is correct (and saved)."
For additional help, you can use the "Step-By](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc167378a-6772-4355-bdb0-a5fec1af8d0e%2F2a4720eb-2832-400c-ac16-046e1d9735cb%2F1gg9zy_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Understanding the Formula for Partial Derivatives**
In this exercise, you are asked to identify the formula for the partial derivative of \( \frac{\partial z}{\partial u} \).
**Options Provided:**
1. \( \frac{\partial z}{\partial x} = \frac{\partial z}{\partial x} \cdot \frac{\partial x}{\partial y} + \frac{\partial z}{\partial y} \cdot \frac{\partial y}{\partial u} \)
2. \( \frac{\partial z}{\partial x} = \frac{\partial z}{\partial x} \cdot \frac{\partial x}{\partial u} + \frac{\partial z}{\partial y} \cdot \frac{\partial y}{\partial u} \)
3. \( \frac{\partial z}{\partial u} = \frac{\partial z}{\partial x} \cdot \frac{\partial x}{\partial u} + \frac{\partial z}{\partial y} \cdot \frac{\partial y}{\partial u} \) (correct answer)
4. \( \frac{\partial z}{\partial x} = \frac{\partial z}{\partial x} \cdot \frac{\partial x}{\partial u} \)
5. \( \frac{\partial z}{\partial y} = \frac{\partial z}{\partial y} \cdot \frac{\partial y}{\partial u} + \frac{\partial z}{\partial u} \cdot \frac{\partial u}{\partial x} \)
**Correct Solution:**
The correct answer is option 3:
\[ \frac{\partial z}{\partial u} = \frac{\partial z}{\partial x} \cdot \frac{\partial x}{\partial u} + \frac{\partial z}{\partial y} \cdot \frac{\partial y}{\partial u} \]
This represents the use of the chain rule in finding the partial derivative of \( z \) with respect to \( u \) by considering how \( z \) depends on intermediate variables \( x \) and \( y \).
**Feedback:**
Once you select the correct formula, you will receive a congratulatory message: "Congratulations, your answer is correct (and saved)."
For additional help, you can use the "Step-By
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