Chapter 12: Q. 25 (page 944)

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

Short Answer

Expert verified

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

Step by step solution

Given

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

To find

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

Calculation

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

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

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

Short Answer

Step by step solution

Given

To find

Calculation

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

Short Answer

Step by step solution

Given

To find

Calculation

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Mechanics Maths

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Pure Maths

Calculus

Study anywhere. Anytime. Across all devices.

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