dQ Use the Chain Rule to find where Q=√√3x² + 3y² + 4z², x= sint, y = cost, and z = cost. 7 dt ƏQ - 2 sin 2t əx √√3+4 3+ 4 cos ²t (Type an expression using x, y, and z as the variables.) dx dt (Type an expression using t as the variable.) ƏQ ду (Type an expression using x, y, and z as the variables.) dy = dt (Type an expression using t as the variable.) ƏQ dz (Type an expression using x, y, and z as the variables.) dz dt ||
dQ Use the Chain Rule to find where Q=√√3x² + 3y² + 4z², x= sint, y = cost, and z = cost. 7 dt ƏQ - 2 sin 2t əx √√3+4 3+ 4 cos ²t (Type an expression using x, y, and z as the variables.) dx dt (Type an expression using t as the variable.) ƏQ ду (Type an expression using x, y, and z as the variables.) dy = dt (Type an expression using t as the variable.) ƏQ dz (Type an expression using x, y, and z as the variables.) dz dt ||
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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Question
![### Chain Rule and Differentiation
**Problem Statement:**
Use the Chain Rule to find \(\frac{dQ}{dt}\), where \(Q = \sqrt{3x^2 + 3y^2 + 4z^2}\), \(x = \sin t\), \(y = \cos t\), and \(z = \cos t\).
**Given Expressions and Definitions:**
\[
\frac{\partial Q}{\partial x} = \boxed{ \frac{-2 \sin 2t}{\sqrt{3 + 4 \cos^2 2t}} } \quad \text{(Type an expression using x, y, and z as the variables.)}
\]
\[
\frac{dx}{dt} = \boxed{ \quad } \quad \text{(Type an expression using t as the variable.)}
\]
\[
\frac{\partial Q}{\partial y} = \boxed{ \quad } \quad \text{(Type an expression using x, y, and z as the variables.)}
\]
\[
\frac{dy}{dt} = \boxed{ \quad } \quad \text{(Type an expression using t as the variable.)}
\]
\[
\frac{\partial Q}{\partial z} = \boxed{ \quad } \quad \text{(Type an expression using x, y, and z as the variables.)}
\]
\[
\frac{dz}{dt} = \boxed{ \quad } \quad \text{(Type an expression using t as the variable.)}
\]
In this problem, we are given a function \(Q\) that depends on three variables \(x\), \(y\), and \(z\), which are themselves functions of \(t\). To find \(\frac{dQ}{dt}\), we need to use the Chain Rule for differentiation, which states that:
\[
\frac{dQ}{dt} = \frac{\partial Q}{\partial x} \frac{dx}{dt} + \frac{\partial Q}{\partial y} \frac{dy}{dt} + \frac{\partial Q}{\partial z} \frac{dz}{dt}
\]
### Detailed Explanation of the Partial Derivatives:
Here are the steps to find each expression:
1. **Find \(\frac{\partial Q}{\](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fdccbb81e-291c-4d6b-9df3-947608720477%2F83f5bd64-f808-4b09-8e31-b16355ae950a%2Fsfjxdpk_processed.png&w=3840&q=75)
Transcribed Image Text:### Chain Rule and Differentiation
**Problem Statement:**
Use the Chain Rule to find \(\frac{dQ}{dt}\), where \(Q = \sqrt{3x^2 + 3y^2 + 4z^2}\), \(x = \sin t\), \(y = \cos t\), and \(z = \cos t\).
**Given Expressions and Definitions:**
\[
\frac{\partial Q}{\partial x} = \boxed{ \frac{-2 \sin 2t}{\sqrt{3 + 4 \cos^2 2t}} } \quad \text{(Type an expression using x, y, and z as the variables.)}
\]
\[
\frac{dx}{dt} = \boxed{ \quad } \quad \text{(Type an expression using t as the variable.)}
\]
\[
\frac{\partial Q}{\partial y} = \boxed{ \quad } \quad \text{(Type an expression using x, y, and z as the variables.)}
\]
\[
\frac{dy}{dt} = \boxed{ \quad } \quad \text{(Type an expression using t as the variable.)}
\]
\[
\frac{\partial Q}{\partial z} = \boxed{ \quad } \quad \text{(Type an expression using x, y, and z as the variables.)}
\]
\[
\frac{dz}{dt} = \boxed{ \quad } \quad \text{(Type an expression using t as the variable.)}
\]
In this problem, we are given a function \(Q\) that depends on three variables \(x\), \(y\), and \(z\), which are themselves functions of \(t\). To find \(\frac{dQ}{dt}\), we need to use the Chain Rule for differentiation, which states that:
\[
\frac{dQ}{dt} = \frac{\partial Q}{\partial x} \frac{dx}{dt} + \frac{\partial Q}{\partial y} \frac{dy}{dt} + \frac{\partial Q}{\partial z} \frac{dz}{dt}
\]
### Detailed Explanation of the Partial Derivatives:
Here are the steps to find each expression:
1. **Find \(\frac{\partial Q}{\
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![### Partial Derivatives and Time Derivatives of \( Q \)
#### Partial Derivative of \( Q \) with Respect to \( x \)
\[
\frac{\partial Q}{\partial x} = \frac{-2 \sin 2t}{\sqrt{3 + 4 \cos^2 2t}}
\]
(*Type an expression using x, y, and z as the variables.*)
#### Time Derivative of \( x \)
\[
\frac{dx}{dt} = \boxed{\text{Type an expression using } t \text{ as the variable.}}
\]
#### Partial Derivative of \( Q \) with Respect to \( y \)
\[
\frac{\partial Q}{\partial y} = \frac{4t \cos t (- \sin t)}{\sqrt{3x^2 + 3y^2 + 4z^2}}
\]
(*Type an expression using x, y, and z as the variables.*)
#### Time Derivative of \( y \)
\[
\frac{dy}{dt} = \frac{3 \sin t \cos t}{\sqrt{3 \sin t^2 + 3 \cos t^2 + 4 \cos 2_t}}
\]
(*Type an expression using t as the variable.*)
#### Partial Derivative of \( Q \) with Respect to \( z \)
\[
\frac{\partial Q}{\partial z} = \frac{3 (\cos t)(- \sin t)}{\sqrt{3 \sin t^2 + 3 \cos t^2 + 4 \cos 2_t}}
\]
(*Type an expression using x, y, and z as the variables.*)
#### Time Derivative of \( z \)
\[
\frac{dz}{dt} = \frac{4 \cos t (- \sin t)}{\sqrt{3 \sin t^2 + 3 \cos t^2 + 4 \cos 2_t}}
\]
(*Type an expression using t as the variable.*)
#### Time Derivative of \( Q \)
\[
\frac{dQ}{dt} = \frac{- 4 \cos t \sin t}{\sqrt{3 + 4 \cos 2_t}}
\]
(*Type an expression using t as the variable.*)
### Explanation of Equations
1. **Partial Derivative](https://content.bartleby.com/qna-images/question/dccbb81e-291c-4d6b-9df3-947608720477/5dfedf5b-9e6c-4af6-99a9-517862e00b95/gjqemu_processed.png)
Transcribed Image Text:### Partial Derivatives and Time Derivatives of \( Q \)
#### Partial Derivative of \( Q \) with Respect to \( x \)
\[
\frac{\partial Q}{\partial x} = \frac{-2 \sin 2t}{\sqrt{3 + 4 \cos^2 2t}}
\]
(*Type an expression using x, y, and z as the variables.*)
#### Time Derivative of \( x \)
\[
\frac{dx}{dt} = \boxed{\text{Type an expression using } t \text{ as the variable.}}
\]
#### Partial Derivative of \( Q \) with Respect to \( y \)
\[
\frac{\partial Q}{\partial y} = \frac{4t \cos t (- \sin t)}{\sqrt{3x^2 + 3y^2 + 4z^2}}
\]
(*Type an expression using x, y, and z as the variables.*)
#### Time Derivative of \( y \)
\[
\frac{dy}{dt} = \frac{3 \sin t \cos t}{\sqrt{3 \sin t^2 + 3 \cos t^2 + 4 \cos 2_t}}
\]
(*Type an expression using t as the variable.*)
#### Partial Derivative of \( Q \) with Respect to \( z \)
\[
\frac{\partial Q}{\partial z} = \frac{3 (\cos t)(- \sin t)}{\sqrt{3 \sin t^2 + 3 \cos t^2 + 4 \cos 2_t}}
\]
(*Type an expression using x, y, and z as the variables.*)
#### Time Derivative of \( z \)
\[
\frac{dz}{dt} = \frac{4 \cos t (- \sin t)}{\sqrt{3 \sin t^2 + 3 \cos t^2 + 4 \cos 2_t}}
\]
(*Type an expression using t as the variable.*)
#### Time Derivative of \( Q \)
\[
\frac{dQ}{dt} = \frac{- 4 \cos t \sin t}{\sqrt{3 + 4 \cos 2_t}}
\]
(*Type an expression using t as the variable.*)
### Explanation of Equations
1. **Partial Derivative
Solution
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