Q 46. In Exercises 43-50, compute all ... [FREE SOLUTION]

Chapter 12: Q 46. (page 944)

In Exercises $43 - 50$ , compute all of the second-order partial derivatives for the functions $g (x, y) = \frac{\ln (y^{2} + 1)}{x}$ and show that the mixed partial derivatives are equal.

Short Answer

Expert verified

The second order partial derivatives for the function are

$\frac{{鈭�}^{2} g}{鈭� x^{2}} = \frac{2 \ln (y^{2} + 1)}{x^{3}} \frac{{鈭�}^{2} g}{鈭� y^{2}} = \frac{4 (1 - y^{2})}{x {(y^{2} + 1)}^{2}}$

Step by step solution

Step 1. Definition

Clairaut's theorem on equality of mixed partials states that under assumption of continuity of both the second-order mixed partials of a function of two variables, the two mixed partials are equal.

Step 2. Finding second order partial derivative

$g (x, y) = \frac{\ln (y^{2} + 1)}{x} 鈰� 鈰� (1)$
Partially differentiate equation $(1)$ both sides with respect to $x$

$鈬� \frac{鈭� g}{鈭� x} = - \frac{\ln (y^{2} + 1)}{x^{2}}$

Again partially differentiate both sides with respect to $x$

$鈬� \frac{{鈭�}^{2} g}{鈭� x^{2}} = \frac{2 \ln (y^{2} + 1)}{x^{3}}$

Partially differentiate equation $(1)$ both sides with respect to $y$

$鈬� \frac{鈭� g}{鈭� y} = \frac{2 y}{x (y^{2} + 1)}$

Again partially differentiate both sides with respect to $y$

$鈬� \frac{{鈭�}^{2} g}{鈭� y^{2}} = \frac{4 (1 - y^{2})}{x {(y^{2} + 1)}^{2}}$

Step 3. Finding mixed order partial derivative

$g (x, y) = \frac{\ln (y^{2} + 1)}{x}$

$鈬� \frac{{鈭�}^{2} g}{鈭� x 鈭� y} = \frac{鈭�}{鈭� x} (\frac{2 y}{x (y^{2} + 1)}) 鈬� \frac{{鈭�}^{2} g}{鈭� x 鈭� y} = - \frac{2 y}{x^{2} (y^{2} + 1)}$

Also

$鈬� \frac{{鈭�}^{2} g}{鈭� y 鈭� x} = \frac{鈭�}{鈭� y} (- \frac{\ln (y^{2} + 1)}{x^{2}}) 鈬� \frac{{鈭�}^{2} g}{鈭� y 鈭� x} = - \frac{2 y}{x^{2} (y^{2} + 1)}$

Now as we observe

$\frac{{鈭�}^{2} g}{鈭� x 鈭� y} = \frac{{鈭�}^{2} g}{鈭� y 鈭� x}$ [Hence proved]

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In Exercises $43 - 50$ , compute all of the second-order partial derivatives for the functions $g (x, y) = \frac{\ln (y^{2} + 1)}{x}$ and show that the mixed partial derivatives are equal.

Short Answer

Step by step solution

Step 1. Definition

Step 2. Finding second order partial derivative

Step 3. Finding mixed order partial derivative

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91影视

In Exercises 43-50, compute all of the second-order partial derivatives for the functions gx,y=lny2+1xand show that the mixed partial derivatives are equal.

Short Answer

Step by step solution

Step 1. Definition

Step 2. Finding second order partial derivative

Step 3. Finding mixed order partial derivative

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Pure Maths

Discrete Mathematics

Geometry

Mechanics Maths

Decision Maths

Study anywhere. Anytime. Across all devices.

In Exercises $43 - 50$ , compute all of the second-order partial derivatives for the functions $g (x, y) = \frac{\ln (y^{2} + 1)}{x}$ and show that the mixed partial derivatives are equal.