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...
Related questions
Question
Find
gx
and gy
![Here is a transcription suitable for an educational website:
---
**Problem Statement:**
Consider the function \( g(x, y) = 0.25x^4 + 3x^2y^{1/3} \). Find the following partial derivatives.
---
In this problem, we are given a multivariable function \( g(x, y) \) involving both \( x \) and \( y \) and are tasked with finding its partial derivatives. The function is composed of two terms: \( 0.25x^4 \) and \( 3x^2y^{1/3} \).
### Step-by-Step Solution:
To solve for the partial derivatives, we will need to apply the rules of differentiation for each variable while treating the other variable as a constant.
1. **Partial derivative with respect to \( x \):**
\[
\frac{\partial g}{\partial x}
\]
2. **Partial derivative with respect to \( y \):**
\[
\frac{\partial g}{\partial y}
\]
We can calculate each of these derivatives by applying the power rule.
---
This problem allows students to practice taking partial derivatives, an essential concept in multivariable calculus. Through solving this, students learn to handle functions of more than one variable, highlighting the utility of calculus in multiple dimensions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0d56d87b-1ee4-40b0-9831-f06655864a4b%2F7cebc660-e249-4066-907b-f9e85e4d6f1e%2Fpj9cixl_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Here is a transcription suitable for an educational website:
---
**Problem Statement:**
Consider the function \( g(x, y) = 0.25x^4 + 3x^2y^{1/3} \). Find the following partial derivatives.
---
In this problem, we are given a multivariable function \( g(x, y) \) involving both \( x \) and \( y \) and are tasked with finding its partial derivatives. The function is composed of two terms: \( 0.25x^4 \) and \( 3x^2y^{1/3} \).
### Step-by-Step Solution:
To solve for the partial derivatives, we will need to apply the rules of differentiation for each variable while treating the other variable as a constant.
1. **Partial derivative with respect to \( x \):**
\[
\frac{\partial g}{\partial x}
\]
2. **Partial derivative with respect to \( y \):**
\[
\frac{\partial g}{\partial y}
\]
We can calculate each of these derivatives by applying the power rule.
---
This problem allows students to practice taking partial derivatives, an essential concept in multivariable calculus. Through solving this, students learn to handle functions of more than one variable, highlighting the utility of calculus in multiple dimensions.
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