Q. 25 Use the definition of the partia... [FREE SOLUTION]

Chapter 12: Q. 25 (page 944)

Use the definition of the partial derivative to find the partial derivatives specified in Exercises 23鈥�26.
$\frac{鈭� f}{鈭� x}, \frac{鈭� f}{鈭� y} and \frac{鈭� f}{鈭� z} when f (x, y, z) = \frac{x y^{2}}{z}$

Short Answer

Expert verified

Required partial derivatives are: $\frac{鈭� f}{鈭� x} = \frac{y^{2}}{z}, \frac{鈭� f}{鈭� y} = \frac{2 x y}{x} and \frac{鈭� f}{鈭� z} = \frac{- x y^{2}}{z^{2}}$

Step by step solution

Given

The given function is: $f (x, y, z) = \frac{x y^{2}}{z}$ .

To find

We have to find $\frac{鈭� f}{鈭� x}, \frac{鈭� f}{鈭� y} and \frac{鈭� f}{鈭� z} .$

Calculation

Differentiate both sides with respect to x, y, and z respectively.

$f (x, y, z) = \frac{x y^{2}}{z} \frac{鈭� f}{鈭� x} = \frac{鈭�}{鈭� x} (\frac{x y^{2}}{z}) \frac{鈭� f}{鈭� x} = \frac{y^{2}}{z} f (x, y, z) = \frac{x y^{2}}{z} \frac{鈭� f}{鈭� y} = \frac{鈭�}{鈭� y} (\frac{x y^{2}}{z}) \frac{鈭� f}{鈭� y} = \frac{2 x y}{x} f (x, y, z) = \frac{x y^{2}}{z} \frac{鈭� f}{鈭� z} = \frac{鈭�}{鈭� z} (\frac{x y^{2}}{z}) \frac{鈭� f}{鈭� z} = \frac{鈭�}{鈭� z} (x y^{2} z^{- 1}) \frac{鈭� f}{鈭� z} = x y^{2} (- 1) z^{- 2} \frac{鈭� f}{鈭� z} = \frac{- x y^{2}}{z^{2}} Hence, \frac{鈭� f}{鈭� x} = \frac{y^{2}}{z}, \frac{鈭� f}{鈭� y} = \frac{2 x y}{x} and \frac{鈭� f}{鈭� z} = \frac{- x y^{2}}{z^{2}}$

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91影视

Use the definition of the partial derivative to find the partial derivatives specified in Exercises 23鈥�26.
$\frac{鈭� f}{鈭� x}, \frac{鈭� f}{鈭� y} and \frac{鈭� f}{鈭� z} when f (x, y, z) = \frac{x y^{2}}{z}$

Short Answer

Step by step solution

Given

To find

Calculation

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91影视

Use the definition of the partial derivative to find the partial derivatives specified in Exercises 23鈥�26. 鈭�f鈭�x,鈭�f鈭�yand鈭�f鈭�zwhenfx,y,z=xy2z

Short Answer

Step by step solution

Given

To find

Calculation

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Logic and Functions

Calculus

Geometry

Discrete Mathematics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Use the definition of the partial derivative to find the partial derivatives specified in Exercises 23鈥�26.
$\frac{鈭� f}{鈭� x}, \frac{鈭� f}{鈭� y} and \frac{鈭� f}{鈭� z} when f (x, y, z) = \frac{x y^{2}}{z}$