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Chapter 12: Q. 60. (page 945)

For each pair of functions in Exercises 59鈥�62, use Theorem
12.24 to show that there is a function of two variables,
$F (x, y)$ such that $\frac{d F}{d x} = g (x, y)$ and $\frac{d F}{d y} = h (x, y)$ Then find $F$ .
$g = \frac{1}{x y} - \sin x, h (x, y) = - \frac{\ln x}{y^{2}}$

Short Answer

Expert verified

The required answer is $F (x, y) = \frac{\ln x}{y} + \cos x + C$

Step by step solution

01

Given information

Think about,

$g = \frac{1}{x y} - \sin x, h = - \frac{\ln x}{y^{2}}$

Then,

$g_{y} = - \frac{1}{x y^{2}} = h_{x}$

There is a function $F (x, y)$ based on the Theorem $12.24$

02

The objective is to find F integration gwith respect to x

Think about,

$鈭� (\frac{1}{x y} - \sin x) d x = \frac{\ln x}{y} + \cos x + q (y) = F (x, y)$

Think about,

$\frac{d}{d y} (F (x, y)) = \frac{d}{d y} (\frac{\ln x}{y} + \cos x + q (y)) = - \frac{\ln x}{y^{2}} + q^{'} (y)$

Suppose,

$\frac{d}{d y} (F (x, y)) = h 鈬� - \frac{\ln x}{y^{2}} + q^{'} (y) = - \frac{\ln x}{y^{2}} q^{'} (y) = 0 q (y) = C$

Hence, $F (x, y) = \frac{\ln x}{y} + \cos x + C$

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Most popular questions from this chapter

Partial derivatives: Find all first- and second-order partial derivatives for the following functions:

$f (x, y, z) = \ln 鈦� (x + y + z)$

Use Theorem 12.33 to find the indicated derivatives in Exercises 27鈥�30. Express your answers as functions of two variables.

$\frac{鈭� z}{鈭� t} w h e n z = x^{2} y^{3}, x = t \sin s, a n d y = s \cos t$

Use Theorem 12.32 to find the indicated derivatives in Exercises 21鈥�26. Express your answers as functions of a single variable.

$\frac{d z}{d t} w h e n z = x^{3} e^{y}, x = \sin t, a n d y = \cos t$

Evaluate the following limits, or explain why the limit does not exist.

$\lim_{(x, y) 鈫� (0, 0)} \frac{x + y}{x^{2} + y^{2}}$

Describe the meanings of each of the following mathematical expressions :

$\begin{aligned} f_{y} (x_{0}, y_{0}) \end{aligned}$

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