**Chain Rule Application to Partial Derivatives** Let \( f(x, y, z) = xy + z^3 \), where \( x = r + s - 5t \), \( y = 3rt \), and \( z = s^4 \). **Task:** Use the Chain Rule to calculate the partial derivatives. *(Use symbolic notation and fractions where needed. Express the answer in terms of independent variables.)* \[ \frac{\partial f}{\partial r} = \text{______} \] \[ \frac{\partial f}{\partial t} = \text{______} \]
**Chain Rule Application to Partial Derivatives** Let \( f(x, y, z) = xy + z^3 \), where \( x = r + s - 5t \), \( y = 3rt \), and \( z = s^4 \). **Task:** Use the Chain Rule to calculate the partial derivatives. *(Use symbolic notation and fractions where needed. Express the answer in terms of independent variables.)* \[ \frac{\partial f}{\partial r} = \text{______} \] \[ \frac{\partial f}{\partial t} = \text{______} \]
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter9: Multivariable Calculus
Section9.CR: Chapter 9 Review
Problem 12CR
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![**Chain Rule Application to Partial Derivatives**
Let \( f(x, y, z) = xy + z^3 \), where \( x = r + s - 5t \), \( y = 3rt \), and \( z = s^4 \).
**Task:** Use the Chain Rule to calculate the partial derivatives.
*(Use symbolic notation and fractions where needed. Express the answer in terms of independent variables.)*
\[
\frac{\partial f}{\partial r} = \text{______}
\]
\[
\frac{\partial f}{\partial t} = \text{______}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F21d63f30-e516-41f1-b813-09fba408cac6%2F13312cd8-3dd5-4bb5-a4c4-a82fc2e70808%2Fjo2z33e_processed.png&w=3840&q=75)
Transcribed Image Text:**Chain Rule Application to Partial Derivatives**
Let \( f(x, y, z) = xy + z^3 \), where \( x = r + s - 5t \), \( y = 3rt \), and \( z = s^4 \).
**Task:** Use the Chain Rule to calculate the partial derivatives.
*(Use symbolic notation and fractions where needed. Express the answer in terms of independent variables.)*
\[
\frac{\partial f}{\partial r} = \text{______}
\]
\[
\frac{\partial f}{\partial t} = \text{______}
\]
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