Q. 28 Find the directional derivative ... [FREE SOLUTION]

Chapter 12: Q. 28 (page 989)

Find the directional derivative of the given function at the specified point P in the direction of the given vector. Note: The given vectors may not be unit vectors.
$f (x, y) = \frac{x y}{x^{2} + y^{2}}, P = (鈭� 2, 1), v = <\frac{3}{5}, 鈭� \frac{4}{5}>$

Short Answer

Expert verified

$D_{v} f (鈭� 2, 1) = \frac{3}{25}$

Step by step solution

Step 1. Given Information

$The directional derivative of a function f (x, y) at a point P = (x_{0}, y_{0}) in the direction of a unit vector u = 鉄� a, b 鉄� is given by D_{u} f (x_{0}, y_{0}) = lim_{h 鈫� 0} 鈥� \frac{f (x_{0} + a h, y_{0} + b h) 鈭� f (x_{0}, y_{0})}{h} Here f (x, y) = \frac{x y}{x^{2} + y^{2}}, P = (鈭� 2, 1) .$

Step 2. Solution

$Given vector is a unit vector so by definition of directional derivative we have: \begin{array}{l} D_{v} f (鈭� 2, 1) = lim_{h 鈫� 0} 鈥� \frac{f (鈭� 2 + \frac{3 h}{5}, 1 鈭� \frac{4 h}{5}) 鈭� f (鈭� 2, 1)}{h} \\ = lim_{h 鈫� 0} 鈥� \frac{\frac{(鈭� 2 + \frac{3 h}{5}) (1 鈭� \frac{4 h}{5})}{{(鈭� 2 + \frac{3 h}{5})}^{2} + {(1 鈭� \frac{4 h}{5})}^{2}} 鈭� \frac{(鈭� 2) (1)}{(鈭� 2)^{2} + (1)^{2}}}{h} \\ = lim_{h 鈫� 0} 鈥� \frac{\frac{鈭� 2 + \frac{11 h}{5} 鈭� \frac{12}{25} h^{2}}{5 鈭� 4 h + h^{2}} 鈭� \frac{2}{5}}{h} \\ Further simplifying the numerator gives: \\ \begin{array}{l} D_{v} f (鈭� 2, 1) = lim_{h 鈫� 0} 鈥� \frac{3 h 鈭� \frac{2}{5} h^{2}}{5 h (5 h 鈭� 4 h + h^{2})} \\ = lim_{h 鈫� 0} 鈥� \frac{3 鈭� \frac{2}{5} h}{5 (5 鈭� 4 h + h^{2})} \\ = \frac{3}{25} \\ H e n c e D_{v} f (鈭� 2, 1) = \frac{3}{25} \end{array} \end{array}$

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

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Find the directional derivative of the given function at the specified point P in the direction of the given vector. Note: The given vectors may not be unit vectors.
$f (x, y) = \frac{x y}{x^{2} + y^{2}}, P = (鈭� 2, 1), v = <\frac{3}{5}, 鈭� \frac{4}{5}>$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Statistics

Geometry

Calculus

Probability and Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

91影视

Find the directional derivative of the given function at the specified point P in the direction of the given vector. Note: The given vectors may not be unit vectors.f(x,y)=xyx2+y2,P=(鈭�2,1),v=35,鈭�45

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Statistics

Geometry

Calculus

Probability and Statistics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Find the directional derivative of the given function at the specified point P in the direction of the given vector. Note: The given vectors may not be unit vectors.
$f (x, y) = \frac{x y}{x^{2} + y^{2}}, P = (鈭� 2, 1), v = <\frac{3}{5}, 鈭� \frac{4}{5}>$