/*! 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 50 Find the first partial derivativ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 12: Problem 50

Find the first partial derivatives of the following functions. $$G(r, s, t)=\sqrt{r s+r t+s t}$$

Short Answer

Expert verified
Question: Find the first partial derivatives of the function $$G(r,s,t) = \sqrt{r s + rt + st}$$. Answer: The first partial derivatives of the function are: - $$\frac{\partial G}{\partial r} = \frac{s+t}{2\sqrt{rs+rt+st}}$$ - $$\frac{\partial G}{\partial s} = \frac{r+t}{2\sqrt{rs+rt+st}}$$ - $$\frac{\partial G}{\partial t} = \frac{r+s}{2\sqrt{rs+rt+st}}$$

Step by step solution

01

Finding the partial derivative with respect to r

To find the partial derivative of G with respect to r, denoted as $$\frac{\partial G}{\partial r}$$, we use the chain rule. The chain rule states: $$\frac{\partial G}{\partial r} = \frac{\partial G}{\partial u} \cdot \frac{\partial u}{\partial r}$$ where u is the expression inside the square root: $$u = r s + rt + st$$. First, find $$\frac{\partial G}{\partial u}$$: $$\frac{\partial G}{\partial u} = \frac{1}{2\sqrt{u}}$$ Next, find $$\frac{\partial u}{\remaining_guesses}$$: $$\frac{\partial u}{\partial r} = s+t$$ Then multiply the two partial derivatives: $$\frac{\partial G}{\partial r} = \frac{1}{2\sqrt{u}}(s+t) = \frac{s+t}{2\sqrt{rs+rt+st}}$$
02

Finding the partial derivative with respect to s

Proceed with finding the partial derivative of G with respect to s, denoted as $$\frac{\partial G}{\partial s}$$, we use the same chain rule as before. Find $$\frac{\partial G}{\partial u}$$: $$\frac{\partial G}{\partial u} = \frac{1}{2\sqrt{u}}$$ Next, find $$\frac{\partial u}{\partial s}$$: $$\frac{\partial u}{\partial s} = r+t$$ Then multiply the two partial derivatives: $$\frac{\partial G}{\partial s} = \frac{1}{2\sqrt{u}}(r+t) = \frac{r+t}{2\sqrt{rs+rt+st}}$$
03

Finding the partial derivative with respect to t

Finally, find the partial derivative of G with respect to t, denoted as $$\frac{\partial G}{\partial t}$$, using the chain rule. Find $$\frac{\partial G}{\partial u}$$: $$\frac{\partial G}{\partial u} = \frac{1}{2\sqrt{u}}$$ Next, find $$\frac{\partial u}{\partial t}$$: $$\frac{\partial u}{\partial t} = r+s$$ Then multiply the two partial derivatives: $$\frac{\partial G}{\partial t} = \frac{1}{2\sqrt{u}}(r+s) = \frac{r+s}{2\sqrt{rs+rt+st}}$$ Now, we have found all three first partial derivatives: - $$\frac{\partial G}{\partial r} = \frac{s+t}{2\sqrt{rs+rt+st}}$$ - $$\frac{\partial G}{\partial s} = \frac{r+t}{2\sqrt{rs+rt+st}}$$ - $$\frac{\partial G}{\partial t} = \frac{r+s}{2\sqrt{rs+rt+st}}$$

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

Consider the ellipse \(x^{2}+4 y^{2}=1\) in the \(x y\) -plane. a. If this ellipse is revolved about the \(x\) -axis, what is the equation of the resulting ellipsoid? b. If this ellipse is revolved about the \(y\) -axis, what is the equation of the resulting ellipsoid?

Find an equation of the plane passing through the point (3,2,1) that slices off the solid in the first octant with the least volume.

The density of a thin circular plate of radius 2 is given by \(\rho(x, y)=4+x y .\) The edge of the plate is described by the parametric equations \(x=2 \cos t, y=2 \sin t,\) for \(0 \leq t \leq 2 \pi\) a. Find the rate of change of the density with respect to \(t\) on the edge of the plate. b. At what point(s) on the edge of the plate is the density a maximum?

A classical equation of mathematics is Laplace's equation, which arises in both theory and applications. It governs ideal fluid flow, electrostatic potentials, and the steadystate distribution of heat in a conducting medium. In two dimensions, Laplace's equation is $$\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0.$$ Show that the following functions are harmonic; that is, they satisfy Laplace's equation. $$u(x, y)=x\left(x^{2}-3 y^{2}\right)$$

Traveling waves (for example, water waves or electromagnetic waves) exhibit periodic motion in both time and position. In one dimension, some types of wave motion are governed by the one-dimensional wave equation $$\frac{\partial^{2} u}{\partial t^{2}}=c^{2} \frac{\partial^{2} u}{\partial x^{2}},$$ where \(u(x, t)\) is the height or displacement of the wave surface at position \(x\) and time \(t,\) and \(c\) is the constant speed of the wave. Show that the following functions are solutions of the wave equation. \(u(x, t)=A f(x+c t)+B g(x-c t),\) where \(A\) and \(B\) are constants and \(f\) and \(g\) are twice differentiable functions of one variable
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Geometry

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation

Probability and Statistics

Read Explanation

Pure Maths

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.