/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 15 Find \(f_{x}\) and \(f_{y}\) $... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 6: Problem 15

Find \(f_{x}\) and \(f_{y}\) $$f(x, y)=x \ln (x y)$$

Short Answer

Expert verified
The partial derivatives are \( f_{x} = \text{ln}(xy) + 1 \) and \( f_{y} = \frac{x}{y} \).

Step by step solution

01

Understand the Problem

We need to find the partial derivatives of the function \( f(x, y) = x \, \text{ln} (xy) \) with respect to both \( x \) and \( y \). Partial derivatives measure how a function changes as one of its variables changes while keeping the other variable constant.
02

Find the Partial Derivative with Respect to \( x \)

To find \( f_{x} \), take the derivative of \( f(x, y) \) with respect to \( x \) while treating \( y \) as a constant. The function can be written as \( f(x, y) = x \, \text{ln}(x) + x \, \text{ln}(y) \). Using the product rule and the chain rule, we get: \( f_{x} = \frac{\text{d}}{\text{d}x}[x \, \text{ln}(xy)] = 1 \, \text{ln}(xy) + x \, \frac{1}{xy} \, y \, \frac{\text{d}}{\text{d}x}[xy] = \text{ln}(xy) + 1 \).
03

Find the Partial Derivative with Respect to \( y \)

To find \( f_{y} \), take the derivative of \( f(x, y) \) with respect to \( y \) while treating \( x \) as a constant. Recall that \( f(x, y) = x \, \text{ln}(xy) \). Using the chain rule, we have: \( f_{y} = \frac{\text{d}}{\text{d}y}[x \, \text{ln}(xy)] = x \, \frac{1}{xy} \, x \cdot \frac{\text{d}}{\text{d}y}[xy] = x \, \frac{1}{xy} \, x = \frac{x}{y} \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Multivariable Calculus
Multivariable calculus generalizes single-variable calculus to functions of more than one variable. This branch encompasses diverse concepts, from partial derivatives to multiple integrals.

Partial derivatives, like \( f_{x} \) and \( f_{y} \) in our problem, highlight how a function changes as only one variable changes, holding the others constant. These derivatives help in understanding the function's behavior in different directions.

Key points about multivariable calculus include:

  • Partial Derivatives: Used to analyze how functions change along specific variables.
  • Gradient: A vector consisting of all partial derivatives, representing the function's greatest rate of increase.
  • Chain and Product Rules: Vital for simplifying the differentiation of composite and multiplied functions.
  • Applications: Extensively used in physics, engineering, economics, and other scientific fields.
Understanding multivariable calculus gives you the tools to navigate complex, real-world problems, providing deep insights into functions with multiple varying factors. Mastering these concepts enables solving detailed and intricate problems across various domains.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Consider \(f(x, y)=x^{3}-5 x y^{2}\). Show that \(f\) is a solution to the partial differential equation \(x f_{x y}-f_{y}=0\).

For data set, find the regression line without using a calculator. $$\begin{array}{|c|c|c|c|}\hline x & 1 & 2 & 4 \\\\\hline y & 3 & 5 & 8 \\\\\hline\end{array}$$

$$\text {Find } f_{x x}, f_{x y}, f_{y x}, \text { and } f_{y y}$$ $$f(x, y)=\frac{x}{y^{2}}-\frac{y}{x^{2}}$$

Use a 3D graphics program to graph each of the following functions. Then estimate any relative extrema. $$f(x, y)=\frac{3 x y\left(x^{2}-y^{2}\right)}{x^{2}+y^{2}}$$

Find the volume of the solid capped by the surface \(z=x+y\) over the region bounded on the \(x y\) -plane by \(y=1-x, y=0, x=0,\) and \(x=1,\) by evaluating the integral $$\int_{0}^{1} \int_{0}^{1-x}(x+y) d y d x$$.
See all solutions

Recommended explanations on Math Textbooks

Statistics

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.