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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![**Instruction for Calculating Partial Derivatives**
Consider the equation:
\[ 5xyz = e^z. \]
We are tasked with finding the partial derivatives \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\). The goal is to express your answers in terms of \(x\), \(y\), and \(z\).
### Steps for Calculation:
1. **Partial Derivative with respect to \(x\):**
To find \(\frac{\partial z}{\partial x}\), differentiate both sides of the equation with respect to \(x\), treating \(y\) and \(z\) as constants where necessary.
\[
\frac{\partial}{\partial x}(5xyz) = \frac{\partial}{\partial x}(e^z).
\]
Fill in your calculated expression for \(\frac{\partial z}{\partial x}\) here:
\[
\frac{\partial z}{\partial x} = \boxed{\ }
\]
2. **Partial Derivative with respect to \(y\):**
Similarly, find \(\frac{\partial z}{\partial y}\) by differentiating with respect to \(y\), treating \(x\) and \(z\) as constants where necessary.
\[
\frac{\partial}{\partial y}(5xyz) = \frac{\partial}{\partial y}(e^z).
\]
Fill in your calculated expression for \(\frac{\partial z}{\partial y}\) here:
\[
\frac{\partial z}{\partial y} = \boxed{\ }
\]
**Note:** Remember that when differentiating exponential functions, the derivative of \(e^z\) with respect to \(z\) will involve applying the chain rule. Keep \(e^z\) as part of your calculations.
This setup helps you calculate the necessary partial derivatives by understanding the relationship between the variables involved.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa4025e11-5cda-4c1f-884f-3d307eee95ea%2Fbab6dd37-126c-4f76-8bd5-8c789699d4ad%2Fnry8uwl_processed.png&w=3840&q=75)
Transcribed Image Text:**Instruction for Calculating Partial Derivatives**
Consider the equation:
\[ 5xyz = e^z. \]
We are tasked with finding the partial derivatives \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\). The goal is to express your answers in terms of \(x\), \(y\), and \(z\).
### Steps for Calculation:
1. **Partial Derivative with respect to \(x\):**
To find \(\frac{\partial z}{\partial x}\), differentiate both sides of the equation with respect to \(x\), treating \(y\) and \(z\) as constants where necessary.
\[
\frac{\partial}{\partial x}(5xyz) = \frac{\partial}{\partial x}(e^z).
\]
Fill in your calculated expression for \(\frac{\partial z}{\partial x}\) here:
\[
\frac{\partial z}{\partial x} = \boxed{\ }
\]
2. **Partial Derivative with respect to \(y\):**
Similarly, find \(\frac{\partial z}{\partial y}\) by differentiating with respect to \(y\), treating \(x\) and \(z\) as constants where necessary.
\[
\frac{\partial}{\partial y}(5xyz) = \frac{\partial}{\partial y}(e^z).
\]
Fill in your calculated expression for \(\frac{\partial z}{\partial y}\) here:
\[
\frac{\partial z}{\partial y} = \boxed{\ }
\]
**Note:** Remember that when differentiating exponential functions, the derivative of \(e^z\) with respect to \(z\) will involve applying the chain rule. Keep \(e^z\) as part of your calculations.
This setup helps you calculate the necessary partial derivatives by understanding the relationship between the variables involved.
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