/*! 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 45 Find the first partial derivativ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 45

Find the first partial derivatives of the following functions. $$F(u, v, w)=\frac{u}{v+w}$$

Short Answer

Expert verified
Question: Find the first partial derivatives of the function F(u, v, w) = u / (v + w). Answer: The first partial derivatives of the function F(u, v, w) with respect to u, v, and w are: $$\frac{\partial F}{\partial u}=\frac{1}{v+w}$$ $$\frac{\partial F}{\partial v}=-\frac{u}{(v+w)^2}$$ $$\frac{\partial F}{\partial w}=-\frac{u}{(v+w)^2}$$

Step by step solution

01

Calculate the Partial Derivative with respect to u

To find the partial derivative of the function F(u,v,w) with respect to u, we will treat v and w as constants and differentiate F with respect to u. The partial derivative with respect to u is denoted as: $$\frac{\partial F}{\partial u}$$ The function we have is: $$F(u, v, w)=\frac{u}{v+w}$$ Applying the derivative with respect to u, we have: $$\frac{\partial F}{\partial u}=\frac{1}{v+w}$$
02

Calculate the Partial Derivative with respect to v

Now, we will find the partial derivative of the function F(u,v,w) with respect to v by treating u and w as constants. The partial derivative with respect to v is denoted as: $$\frac{\partial F}{\partial v}$$ The function is still: $$F(u, v, w)=\frac{u}{v+w}$$ Applying the derivative with respect to v, we have: $$\frac{\partial F}{\partial v}=-\frac{u}{(v+w)^2}$$
03

Calculate the Partial Derivative with respect to w

Lastly, we will find the partial derivative of the function F(u,v,w) with respect to w, while treating u and v as constants. The partial derivative with respect to w is denoted as: $$\frac{\partial F}{\partial w}$$ The function remains: $$F(u, v, w)=\frac{u}{v+w}$$ Applying the derivative with respect to w, we have: $$\frac{\partial F}{\partial w}=-\frac{u}{(v+w)^2}$$
04

Final Result

The first partial derivatives of the function F(u, v, w) with respect to u, v, and w are as follows: $$\frac{\partial F}{\partial u}=\frac{1}{v+w}$$ $$\frac{\partial F}{\partial v}=-\frac{u}{(v+w)^2}$$ $$\frac{\partial F}{\partial w}=-\frac{u}{(v+w)^2}$$

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!

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

Use the definition of the gradient (in two or three dimensions), assume that \(f\) and \(g\) are differentiable functions on \(\mathbb{R}^{2}\) or \(\mathbb{R}^{3},\) and let \(c\) be a constant. Prove the following gradient rules. a. Constants Rule: \(\nabla(c f)=c \nabla f\) b. Sum Rule: \(\nabla(f+g)=\nabla f+\nabla g\) c. Product Rule: \(\nabla(f g)=(\nabla f) g+f \nabla g\) d. Quotient Rule: \(\nabla\left(\frac{f}{g}\right)=\frac{g \nabla f-f \nabla g}{g^{2}}\) e. Chain Rule: \(\nabla(f \circ g)=f^{\prime}(g) \nabla g,\) where \(f\) is a function of one variable

Use the method of your choice to ate the following limits. $$\lim _{(x, y) \rightarrow(2,0)} \frac{1-\cos y}{x y^{2}}$$

Identify and briefly describe the surfaces defined by the following equations. $$-y^{2}-9 z^{2}+x^{2} / 4=1$$

The flow of heat along a thin conducting bar is governed by the one- dimensional heat equation (with analogs for thin plates in two dimensions and for solids in three dimensions) $$\frac{\partial u}{\partial t}=k \frac{\partial^{2} u}{\partial x^{2}},$$ where \(u\) is a measure of the temperature at a location \(x\) on the bar at time t and the positive constant \(k\) is related to the conductivity of the material. Show that the following functions satisfy the heat equation with \(k=1.\) $$u(x, t)=e^{-t}(2 \sin x+3 \cos x)$$

Production functions Economists model the output of manufacturing systems using production functions that have many of the same properties as utility functions. The family of Cobb-Douglas production functions has the form \(P=f(K, L)=C K^{a} L^{1-a},\) where K represents capital, L represents labor, and C and a are positive real numbers with \(0
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Calculus

Read Explanation

Statistics

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.