### Calculating Partial Derivatives and Applying the Chain Rule Given: \[ f(x, y, z) = x^5 y^5 + z^5 \] \[ x = s^5 \] \[ y = s t^4 \] \[ z = s^5 t \] ### (a) Calculate the primary derivatives \(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\) (Use symbolic notation and fractions where needed.) \[ \frac{\partial f}{\partial x} = 5x^4 y^5 \] \[ \frac{\partial f}{\partial y} = 5x^5 y^4 \] \[ \frac{\partial f}{\partial z} = 5z^4 \] ### (b) Calculate \(\frac{\partial x}{\partial s}, \frac{\partial y}{\partial s}, \frac{\partial z}{\partial s}\) (Use symbolic notation and fractions where needed.) \[ \frac{\partial x}{\partial s} = 5s^4 \] \[ \frac{\partial y}{\partial s} = t^4 \] \[ \frac{\partial z}{\partial s} = 5s^4 t \] ### (c) Compute \(\frac{\partial f}{\partial s}\) using the Chain Rule. \[ \frac{\partial f}{\partial s} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial s} + \frac{\partial f}{\partial z} \frac{\partial z}{\partial s} \] (Use symbolic notation and fractions where needed.) \[ \frac{\partial f}{\partial s} = \]
### Calculating Partial Derivatives and Applying the Chain Rule Given: \[ f(x, y, z) = x^5 y^5 + z^5 \] \[ x = s^5 \] \[ y = s t^4 \] \[ z = s^5 t \] ### (a) Calculate the primary derivatives \(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\) (Use symbolic notation and fractions where needed.) \[ \frac{\partial f}{\partial x} = 5x^4 y^5 \] \[ \frac{\partial f}{\partial y} = 5x^5 y^4 \] \[ \frac{\partial f}{\partial z} = 5z^4 \] ### (b) Calculate \(\frac{\partial x}{\partial s}, \frac{\partial y}{\partial s}, \frac{\partial z}{\partial s}\) (Use symbolic notation and fractions where needed.) \[ \frac{\partial x}{\partial s} = 5s^4 \] \[ \frac{\partial y}{\partial s} = t^4 \] \[ \frac{\partial z}{\partial s} = 5s^4 t \] ### (c) Compute \(\frac{\partial f}{\partial s}\) using the Chain Rule. \[ \frac{\partial f}{\partial s} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial s} + \frac{\partial f}{\partial z} \frac{\partial z}{\partial s} \] (Use symbolic notation and fractions where needed.) \[ \frac{\partial f}{\partial s} = \]
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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I need help with part C
![### Calculating Partial Derivatives and Applying the Chain Rule
Given:
\[ f(x, y, z) = x^5 y^5 + z^5 \]
\[ x = s^5 \]
\[ y = s t^4 \]
\[ z = s^5 t \]
### (a) Calculate the primary derivatives \(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\)
(Use symbolic notation and fractions where needed.)
\[
\frac{\partial f}{\partial x} = 5x^4 y^5
\]
\[
\frac{\partial f}{\partial y} = 5x^5 y^4
\]
\[
\frac{\partial f}{\partial z} = 5z^4
\]
### (b) Calculate \(\frac{\partial x}{\partial s}, \frac{\partial y}{\partial s}, \frac{\partial z}{\partial s}\)
(Use symbolic notation and fractions where needed.)
\[
\frac{\partial x}{\partial s} = 5s^4
\]
\[
\frac{\partial y}{\partial s} = t^4
\]
\[
\frac{\partial z}{\partial s} = 5s^4 t
\]
### (c) Compute \(\frac{\partial f}{\partial s}\) using the Chain Rule.
\[
\frac{\partial f}{\partial s} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial s} + \frac{\partial f}{\partial z} \frac{\partial z}{\partial s}
\]
(Use symbolic notation and fractions where needed.)
\[
\frac{\partial f}{\partial s} =
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faccc7427-2c55-44a5-b68e-19581b7568be%2Ffa7228e4-8c23-429e-8eab-0592543af8c2%2Fcl0x9g_processed.png&w=3840&q=75)
Transcribed Image Text:### Calculating Partial Derivatives and Applying the Chain Rule
Given:
\[ f(x, y, z) = x^5 y^5 + z^5 \]
\[ x = s^5 \]
\[ y = s t^4 \]
\[ z = s^5 t \]
### (a) Calculate the primary derivatives \(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z}\)
(Use symbolic notation and fractions where needed.)
\[
\frac{\partial f}{\partial x} = 5x^4 y^5
\]
\[
\frac{\partial f}{\partial y} = 5x^5 y^4
\]
\[
\frac{\partial f}{\partial z} = 5z^4
\]
### (b) Calculate \(\frac{\partial x}{\partial s}, \frac{\partial y}{\partial s}, \frac{\partial z}{\partial s}\)
(Use symbolic notation and fractions where needed.)
\[
\frac{\partial x}{\partial s} = 5s^4
\]
\[
\frac{\partial y}{\partial s} = t^4
\]
\[
\frac{\partial z}{\partial s} = 5s^4 t
\]
### (c) Compute \(\frac{\partial f}{\partial s}\) using the Chain Rule.
\[
\frac{\partial f}{\partial s} = \frac{\partial f}{\partial x} \frac{\partial x}{\partial s} + \frac{\partial f}{\partial y} \frac{\partial y}{\partial s} + \frac{\partial f}{\partial z} \frac{\partial z}{\partial s}
\]
(Use symbolic notation and fractions where needed.)
\[
\frac{\partial f}{\partial s} =
\]
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