Q. 54. Use Theorem 12.32 to find the in... [FREE SOLUTION]

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Chapter 12: Q. 54. (page 965)

Use Theorem 12.32 to find the indicated derivatives in Exercises
21鈥�26. Express your answers as functions of a single variable
$w = \ln (\frac{x^{2}}{y z}) + \ln (\frac{z}{x y}) - \ln (\frac{x}{y^{2}}), P (3, 5, 8), \overset{鈫�}{v} = 13 i + 21 j + 34 k$

Short Answer

Expert verified

The direction derivative of the given function at $P (3, 5, 8)$ in the direction $\overset{鈫�}{v}$

role="math" localid="1653917061156" $D_{\overset{鈫�}{v}} w (x, y, z) = 0$

Step by step solution

Given information

The function $w = \ln (\frac{x^{2}}{y z}) + \ln (\frac{z}{x y}) - \ln (\frac{x}{y^{2}}) 鈥� 鈥� . (1)$

And the given vector

$\overset{鈫�}{v} = 13 i + 21 j + 34 k$

The objective is to find the direct derivative of the given function atP(3,5,8) in the direction v→

The direction derivative of a function $f (x, y, z)$ in the direction of the unit vector $\overset{鈫�}{u} = a i + b j + c j$

$D_{u} (x, y, z) = f_{x} (x, y, z) a + f_{y} (x, y, z) b + f_{z} (x, y, z) c = 鈭� f (x, y, z) 路 \overset{鈫�}{u}$

Rewrite function $(1)$ as follows:

$w = \ln (\frac{x^{2}}{y z}) + \ln (\frac{z}{x y}) - \ln (\frac{x}{y^{2}}) = \ln (\frac{x^{2}}{y z}) (\frac{z}{x y}) - \ln (\frac{x}{y^{2}}) = \ln (\frac{x}{y^{2}}) (\frac{y^{2}}{x}) = \ln 1 = 0 T h u s, 鈭� w = 0$

Hence, the directional derivative of the given function at $P (3, 5, 8)$ in the direction $\overset{鈫�}{v}$

$D_{\overset{鈫�}{v}} w (x, y, z) = 鈭� w (x, y, z) 路 \overset{鈫�}{v} = 0 路 \overset{鈫�}{v} = 0$

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91影视

Use Theorem 12.32 to find the indicated derivatives in Exercises
21鈥�26. Express your answers as functions of a single variable
$w = \ln (\frac{x^{2}}{y z}) + \ln (\frac{z}{x y}) - \ln (\frac{x}{y^{2}}), P (3, 5, 8), \overset{鈫�}{v} = 13 i + 21 j + 34 k$

Short Answer

Step by step solution

Given information

The objective is to find the direct derivative of the given function atP(3,5,8) in the direction v→

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91影视

Use Theorem 12.32 to find the indicated derivatives in Exercises21鈥�26. Express your answers as functions of a single variablew=lnx2yz+lnzxy-lnxy2,P(3,5,8),v鈫�=13i+21j+34k

Short Answer

Step by step solution

Given information

The objective is to find the direct derivative of the given function atP(3,5,8) in the direction v→

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Mechanics Maths

Statistics

Decision Maths

Theoretical and Mathematical Physics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Use Theorem 12.32 to find the indicated derivatives in Exercises
21鈥�26. Express your answers as functions of a single variable
$w = \ln (\frac{x^{2}}{y z}) + \ln (\frac{z}{x y}) - \ln (\frac{x}{y^{2}}), P (3, 5, 8), \overset{鈫�}{v} = 13 i + 21 j + 34 k$