Chapter 12: Q. 25 (page 953)

In Exercises $21 - 28$ , find the directional derivative of the given function at the specified point $P$ and in the direction of the given unit vector $u$ .
$f (x, y) = \sqrt{\frac{y}{x}}$ at $P = (4, 9), u = (- \frac{\sqrt{17}}{17}, - \frac{4 \sqrt{17}}{17})$

Short Answer

Expert verified

Directional derivative for the given function is $\nabla f (P) \times u = - \frac{7}{816} \sqrt{17}$ .

Step by step solution

Given information

Directional derivative of function is,

$f (x, y) = \sqrt{\frac{y}{x}}$

Given is,

$P = (x_{0}, y_{0}) = (4, 9)$ and $u = (α, β) = (- \frac{\sqrt{17}}{17}, - \frac{4 \sqrt{17}}{17})$

Calculation for directional derivative

Direction derivative of function,

$\nabla f (P) \times u = \nabla f (4, 9) \times u$

$= ({\frac{df}{dx}|}_{(4, 9)} i + {\frac{df}{dy}|}_{(4, 9)} j) \times (- \frac{\sqrt{17}}{17} i - \frac{4 \sqrt{17}}{17} j)$

$= [{(\frac{1}{2 \sqrt{\frac{y}{x}}} \frac{d}{dx} (\frac{y}{x}))}_{(4, 9)} i + {(\frac{1}{2 \sqrt{\frac{y}{x}}} \frac{d}{dx} (\frac{y}{x}))}_{(4, 9)} j] \times (- \frac{\sqrt{17}}{17} i - \frac{4 \sqrt{17}}{17} j)$

$= [{(\frac{\frac{- y}{x^{2}}}{2 \sqrt{\frac{y}{x}}})}_{(4, 9)} i + {(\frac{\frac{1}{x}}{2 \sqrt{\frac{y}{x}}})}_{(4, 9)} j] \times (- \frac{\sqrt{17}}{17} i - \frac{4 \sqrt{17}}{17} j)$

$= [(\frac{\frac{- 9}{4^{2}}}{2 \sqrt{\frac{9}{4}}}) i + (\frac{\frac{1}{4}}{2 \sqrt{\frac{9}{4}}}) j] \times (- \frac{\sqrt{17}}{17} i - \frac{4 \sqrt{17}}{17} j)$

$= [(\frac{- \frac{9}{16}}{2 \times \frac{3}{2}}) i + (\frac{\frac{1}{4}}{2 \times \frac{3}{2}}) j] \times (- \frac{\sqrt{17}}{17} i - \frac{4 \sqrt{17}}{17} j)$

$= (- \frac{9}{48} i + \frac{1}{12} j) \times (- \frac{\sqrt{17}}{17} i - \frac{4 \sqrt{17}}{17} j)$

$= (\frac{9 \times \sqrt{17}}{48 \times 17} - \frac{4 \sqrt{17}}{12 \times 17})$

$= (\frac{9 \times \sqrt{17}}{16 \times 3 \times 17} - \frac{\sqrt{17}}{3 \times 17})$

$= \frac{1}{3 \times 17} (\frac{9 \times \sqrt{17}}{16} - \frac{\sqrt{17}}{1})$

$= \frac{1}{3 \times 17} (\frac{9 \times \sqrt{17} - 16 \sqrt{17}}{16})$

role="math" localid="1650565813026" $\nabla f (P) \times u = - \frac{7}{816} \sqrt{17}$

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In Exercises $21 - 28$ , find the directional derivative of the given function at the specified point $P$ and in the direction of the given unit vector $u$ .
$f (x, y) = \sqrt{\frac{y}{x}}$ at $P = (4, 9), u = (- \frac{\sqrt{17}}{17}, - \frac{4 \sqrt{17}}{17})$

Short Answer

Step by step solution

Given information

Calculation for directional derivative

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In Exercises 21-28, find the directional derivative of the given function at the specified point Pand in the direction of the given unit vectoru .f(x,y)=yxat P=(4,9),u=-1717,-41717

Short Answer

Step by step solution

Given information

Calculation for directional derivative

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Mechanics Maths

Geometry

Pure Maths

Decision Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

In Exercises $21 - 28$ , find the directional derivative of the given function at the specified point $P$ and in the direction of the given unit vector $u$ .
$f (x, y) = \sqrt{\frac{y}{x}}$ at $P = (4, 9), u = (- \frac{\sqrt{17}}{17}, - \frac{4 \sqrt{17}}{17})$