Chapter 12: Q. 35 (page 944)

Find the first-order partial derivatives for the functions in Exercises 27–36.
$f (x, y, z) = \frac{x y^{2}}{x + z}$

Short Answer

Expert verified

The first-order partial derivatives are $\frac{\partial f}{\partial x} = \frac{z y^{2}}{{(x + z)}^{2}}, \frac{\partial f}{\partial y} = \frac{2 x y}{x + z}, \frac{\partial f}{\partial z} = - \frac{x y^{2}}{{(x + z)}^{2}} .$

Step by step solution

Given

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

To find

We have to find the first-order partial derivatives.

Calculation

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

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Find the first-order partial derivatives for the functions in Exercises 27–36.
$f (x, y, z) = \frac{x y^{2}}{x + z}$

Short Answer

Step by step solution

Given

To find

Calculation

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Find the first-order partial derivatives for the functions in Exercises 27–36. fx,y,z=xy2x+z

Short Answer

Step by step solution

Given

To find

Calculation

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Most popular questions from this chapter

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Find the first-order partial derivatives for the functions in Exercises 27–36.
$f (x, y, z) = \frac{x y^{2}}{x + z}$