= any Let z(x, y) = dz dt Calculate dx dt dy 9,8 dt 5n -4e* sin(y) where x by first finding & Hint: Recall the chain rule: Question Help: Video dt Now use the chain rule to calculate the following: dz Submit Question AZE dx dy dt dt dz dt tº & y = 5nt. and using the chain rule. Search dx dy (8x de ) + (du) dt dt
= any Let z(x, y) = dz dt Calculate dx dt dy 9,8 dt 5n -4e* sin(y) where x by first finding & Hint: Recall the chain rule: Question Help: Video dt Now use the chain rule to calculate the following: dz Submit Question AZE dx dy dt dt dz dt tº & y = 5nt. and using the chain rule. Search dx dy (8x de ) + (du) dt dt
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Problem Statement:**
Let \( z(x, y) = -4e^x \sin(y) \) where \( x = t^9 \) and \( y = 5\pi t \).
**Objective:**
Calculate \( \frac{dz}{dt} \) by first finding \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \) and using the chain rule.
**Given:**
\[ \frac{dx}{dt} = 9t^8 \]
\[ \frac{dy}{dt} = 5\pi \]
**Task:**
Now use the chain rule to calculate the following:
\[ \frac{dz}{dt} = \]
**Hint:**
Recall the chain rule:
\[ \frac{dz}{dt} = \left( \frac{\partial z}{\partial x} \right) \left( \frac{dx}{dt} \right) + \left( \frac{\partial z}{\partial y} \right) \left( \frac{dy}{dt} \right) \]
**Instructions:**
1. **Calculate Partial Derivatives**:
- Compute \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \).
2. **Apply the Chain Rule**:
- Use the chain rule formula to find \( \frac{dz}{dt} \) by substituting the necessary derivatives.
3. **Use Given Information**:
- Use the calculated values of \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \) to complete the process.
**Assistance:**
For additional help, click on the video icon.
**Submission:**
After completing the calculations, press the "Submit Question" button.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8a3f3121-07ef-4e2d-8122-d3ccd0e5a601%2F42bbb5b0-d8bb-44f9-b7c3-e69e27a5dc10%2Fprmrkj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Let \( z(x, y) = -4e^x \sin(y) \) where \( x = t^9 \) and \( y = 5\pi t \).
**Objective:**
Calculate \( \frac{dz}{dt} \) by first finding \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \) and using the chain rule.
**Given:**
\[ \frac{dx}{dt} = 9t^8 \]
\[ \frac{dy}{dt} = 5\pi \]
**Task:**
Now use the chain rule to calculate the following:
\[ \frac{dz}{dt} = \]
**Hint:**
Recall the chain rule:
\[ \frac{dz}{dt} = \left( \frac{\partial z}{\partial x} \right) \left( \frac{dx}{dt} \right) + \left( \frac{\partial z}{\partial y} \right) \left( \frac{dy}{dt} \right) \]
**Instructions:**
1. **Calculate Partial Derivatives**:
- Compute \( \frac{\partial z}{\partial x} \) and \( \frac{\partial z}{\partial y} \).
2. **Apply the Chain Rule**:
- Use the chain rule formula to find \( \frac{dz}{dt} \) by substituting the necessary derivatives.
3. **Use Given Information**:
- Use the calculated values of \( \frac{dx}{dt} \) and \( \frac{dy}{dt} \) to complete the process.
**Assistance:**
For additional help, click on the video icon.
**Submission:**
After completing the calculations, press the "Submit Question" button.
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