/*! 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 16 Find the first partial derivativ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 16

Find the first partial derivatives of the following functions. $$f(s, t)=\frac{s-t}{s+t}$$

Short Answer

Expert verified
The first partial derivatives are: $$\frac{\partial f}{\partial s}=\frac{2t}{(s+t)^2}$$ $$\frac{\partial f}{\partial t}=\frac{-2s}{(s+t)^2}$$

Step by step solution

01

Review function and differentiation rules

Our function is given as: $$f(s, t)=\frac{s-t}{s+t}$$ We will be using the Leibniz's Notation for finding partial derivatives of a function with respect to each independent variable (s and t). The rules of differentiation for addition, subtraction, and the division rule will be applied.
02

Differentiate with respect to s

We will now find the partial derivative of the function with respect to s. Using the quotient rule of differentiation, we have: $$\frac{\partial f}{\partial s}=\frac{\frac{\partial}{\partial s} (s-t)\cdot(s+t)-\frac{\partial}{\partial s} (s+t)\cdot(s-t)}{(s+t)^2}$$ Now, let's differentiate each term inside the fractions. \(\frac{\partial}{\partial s}(s-t)=1\), \(\frac{\partial}{\partial s}(s+t)=1\): $$\frac{\partial f}{\partial s}=\frac{(1)\cdot(s+t)-(1)\cdot(s-t)}{(s+t)^2}$$ Now, we simplify the numerator: $$\frac{\partial f}{\partial s}=\frac{2t}{(s+t)^2}$$
03

Differentiate with respect to t

We will now find the partial derivative of the function with respect to t. Using the quotient rule of differentiation, we have: $$\frac{\partial f}{\partial t}=\frac{\frac{\partial}{\partial t} (s-t)\cdot(s+t)-\frac{\partial}{\partial t} (s+t)\cdot(s-t)}{(s+t)^2}$$ Now, let's differentiate each term inside the fractions. \(\frac{\partial}{\partial t}(s-t)=-1\), \(\frac{\partial}{\partial t}(s+t)=1\): $$\frac{\partial f}{\partial t}=\frac{(-1)\cdot(s+t)-(1)\cdot(s-t)}{(s+t)^2}$$ Now, we simplify the numerator: $$\frac{\partial f}{\partial t}=\frac{-2s}{(s+t)^2}$$
04

Write the results

Finally, we have both first-order partial derivatives: $$\frac{\partial f}{\partial s}=\frac{2t}{(s+t)^2}$$ $$\frac{\partial f}{\partial t}=\frac{-2s}{(s+t)^2}$$ These are the first partial derivatives of the given function \(f(s, t)=\frac{s-t}{s+t}\).

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

Find an equation of the plane passing through (0,-2,4) that is orthogonal to the planes \(2 x+5 y-3 z=0\) and \(-x+5 y+2 z=8\)

Consider the following equations of quadric surfaces. a. Find the intercepts with the three coordinate axes, when they exist. b. Find the equations of the \(x y-, x z^{-}\), and \(y z\) -traces, when they exist. c. Sketch a graph of the surface. $$-\frac{x^{2}}{3}+3 y^{2}-\frac{z^{2}}{12}=1$$

Absolute maximum and minimum values Find the absolute maximum and minimum values of the following functions over the given regions \(R\). Use Lagrange multipliers to check for extreme points on the boundary. $$f(x, y)=x^{2}+4 y^{2}+1 ; R=\left\\{(x, y): x^{2}+4 y^{2} \leq 1\right\\}$$

Evaluate the following limits. $$\lim _{(x, y) \rightarrow(0, \pi / 2)} \frac{1-\cos x y}{4 x^{2} y^{3}}$$

Evaluate the following limits. $$a.\lim _{(x, y) \rightarrow(0,0)} \frac{\sin (x+y)}{x+y}$$ $$b.\lim _{(x, y) \rightarrow(0,0)} \frac{\sin x+\sin y}{x+y}$$
See all solutions

Recommended explanations on Math Textbooks

Calculus

Read Explanation

Statistics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Probability and 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.