Can someone check my work The image contains two examples of computing the rate of change of a function using the chain rule in multivariable calculus.**Example 1:**Given the function:\[ w = x^3y^2 + x^4y^3z \]- $ x = \cos(4t) $- $ y = \sin(5t) $- $ z = \ln(3t) $**Diagram:**There is a tree diagram showing the dependencies:- The top node is $ w $.- The second layer has nodes for $ x $, $ y $, and $ z $.- The third layer has a node for $ t $.**Chain Rule Calculation:**The derivative $ \frac{dw}{dt} $ is calculated using the partial derivatives and the derivatives of $ x $, $ y $, and $ z $ with respect to $ t $. Each term is expanded and simplified step by step.**Example 2:**Given the function:\[ z = \ln(3x + 2y) \]- $ x = s \cdot \sin(t) $- $ y = 4 \cdot \cos(t) $**Diagram:**A similar dependency tree diagram shows:- The top node is $ z $.- The second layer has nodes for $ x $ and $ y $.- The bottom layer has nodes for $ s $ and $ t $.**Chain Rule Calculation:**The derivative $ \frac{dz}{dt} $ is calculated, involving partial derivatives $ \frac{\partial z}{\partial x} $, $ \frac{\partial z}{\partial y} $, $ \frac{\partial x}{\partial t} $, $ \frac{\partial y}{\partial t} $. The expressions are expanded and simplified to give the final result.**Visual elements highlighted:**- Red for partial derivatives.- Green background for the part of the diagram related to $ \frac{dz}{dt} $.Each mathematical expression is expanded, showing detailed steps for educational purposes, illustrating how to apply the chain rule in complex scenarios involving multiple variables and paths of dependency. 1. **Given** $ w = x^2 y^3 z + x^4 y z^3 $, $ x = \cos(4t) $, $ y = \sin(5t) $, $ z = \ln(t) $. Write the tree diagram for $\frac{dw}{dt}$. Then, find $\frac{dw}{dt}$.2. **Given** $ z = \ln(3x + 2y) $, $ x = s \cdot \sin(t) $, $ y = t \cdot \cos(t) $. Find $\frac{\partial z}{\partial s}$ and $\frac{\partial z}{\partial t}$.

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Description: Can someone check my work The image contains two examples of computing the rate of change of a function using the chain rule in multivariable calculus.**Example 1:**Given the function:\[ w = x^3y^2 + x^4y^3z \]- $ x = \cos(4t) $- $ y = \sin(5t) $

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Answered: 1. Given w = x²y³z+x*yz³, x = cos(4t),y…

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1. Given w = x²y³z+x*yz³, x = cos(4t),y = sin(5t),z = In(t). Write the tree diagram for . Then, find. 2. Given z=ln(3x+2y), x=s•sin(t), y=t• cos(t). Find and . dw dw dt dt əz əz əs at

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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Can someone check my work

The image contains two examples of computing the rate of change of a function using the chain rule in multivariable calculus.

**Example 1:**

Given the function:
\[ w = x^3y^2 + x^4y^3z \]
- $ x = \cos(4t) $
- $ y = \sin(5t) $
- $ z = \ln(3t) $

**Diagram:**
There is a tree diagram showing the dependencies:
- The top node is $ w $.
- The second layer has nodes for $ x $, $ y $, and $ z $.
- The third layer has a node for $ t $.

**Chain Rule Calculation:**
The derivative $ \frac{dw}{dt} $ is calculated using the partial derivatives and the derivatives of $ x $, $ y $, and $ z $ with respect to $ t $. Each term is expanded and simplified step by step.

**Example 2:**

Given the function:
\[ z = \ln(3x + 2y) \]
- $ x = s \cdot \sin(t) $
- $ y = 4 \cdot \cos(t) $

**Diagram:**
A similar dependency tree diagram shows:
- The top node is $ z $.
- The second layer has nodes for $ x $ and $ y $.
- The bottom layer has nodes for $ s $ and $ t $.

**Chain Rule Calculation:**
The derivative $ \frac{dz}{dt} $ is calculated, involving partial derivatives $ \frac{\partial z}{\partial x} $, $ \frac{\partial z}{\partial y} $, $ \frac{\partial x}{\partial t} $, $ \frac{\partial y}{\partial t} $. The expressions are expanded and simplified to give the final result.

**Visual elements highlighted:**
- Red for partial derivatives.
- Green background for the part of the diagram related to $ \frac{dz}{dt} $.

Each mathematical expression is expanded, showing detailed steps for educational purposes, illustrating how to apply the chain rule in complex scenarios involving multiple variables and paths of dependency.

1. **Given** $ w = x^2 y^3 z + x^4 y z^3 $, $ x = \cos(4t) $, $ y = \sin(5t) $, $ z = \ln(t) $. Write the tree diagram for $\frac{dw}{dt}$. Then, find $\frac{dw}{dt}$.

2. **Given** $ z = \ln(3x + 2y) $, $ x = s \cdot \sin(t) $, $ y = t \cdot \cos(t) $. Find $\frac{\partial z}{\partial s}$ and $\frac{\partial z}{\partial t}$.

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