Question 4 Chain Rule (One independent variable) dw Calculate by first finding dt Let w(x, y, z) = x² + y² +2² where x = : dx dt dy dt II || < || > Submit Question dx dy dt 1 sin(5t), y = cos(-3t), z = e²t. dz & dt dt dz dt Now use the chain rule to calculate the following: dw dt and using the chain rule.
Question 4 Chain Rule (One independent variable) dw Calculate by first finding dt Let w(x, y, z) = x² + y² +2² where x = : dx dt dy dt II || < || > Submit Question dx dy dt 1 sin(5t), y = cos(-3t), z = e²t. dz & dt dt dz dt Now use the chain rule to calculate the following: dw dt and using the chain rule.
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
![**Chain Rule (One Independent Variable)**
Let \( w(x, y, z) = x^2 + y^2 + z^2 \) where \( x = \sin(5t) \), \( y = \cos(-3t) \), \( z = e^{2t} \).
Calculate \( \frac{dw}{dt} \) by first finding \( \frac{dx}{dt} \), \( \frac{dy}{dt} \), and \( \frac{dz}{dt} \) and using the chain rule.
\[
\frac{dx}{dt} = \, \text{[blank]}
\]
\[
\frac{dy}{dt} = \, \text{[blank]}
\]
\[
\frac{dz}{dt} = \, \text{[blank]}
\]
Now use the chain rule to calculate the following:
\[
\frac{dw}{dt} = \, \text{[blank]}
\]
---
**Instructions:**
- First, compute each derivative \( \frac{dx}{dt} \), \( \frac{dy}{dt} \), and \( \frac{dz}{dt} \).
- Apply the chain rule to find \( \frac{dw}{dt} \).
- Fill in each box with the appropriate derivative or result.
- Once finished, submit your answers.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8a3f3121-07ef-4e2d-8122-d3ccd0e5a601%2F087df6b0-a8da-4525-b147-df5f417e7561%2F8g5hiqm_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Chain Rule (One Independent Variable)**
Let \( w(x, y, z) = x^2 + y^2 + z^2 \) where \( x = \sin(5t) \), \( y = \cos(-3t) \), \( z = e^{2t} \).
Calculate \( \frac{dw}{dt} \) by first finding \( \frac{dx}{dt} \), \( \frac{dy}{dt} \), and \( \frac{dz}{dt} \) and using the chain rule.
\[
\frac{dx}{dt} = \, \text{[blank]}
\]
\[
\frac{dy}{dt} = \, \text{[blank]}
\]
\[
\frac{dz}{dt} = \, \text{[blank]}
\]
Now use the chain rule to calculate the following:
\[
\frac{dw}{dt} = \, \text{[blank]}
\]
---
**Instructions:**
- First, compute each derivative \( \frac{dx}{dt} \), \( \frac{dy}{dt} \), and \( \frac{dz}{dt} \).
- Apply the chain rule to find \( \frac{dw}{dt} \).
- Fill in each box with the appropriate derivative or result.
- Once finished, submit your answers.
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