/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 86 Use the gradient rules of Exerci... [FREE SOLUTION] | 91影视

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Chapter 13: Problem 86

Use the gradient rules of Exercise 81 to find the gradient of the following functions. $$f(x, y, z)=(x+y+z) e^{x y z}$$

Short Answer

Expert verified
Answer: The gradient of the function $$f(x, y, z) = (x + y + z)e^{xyz}$$ is $$\nabla f = \left( (1+yze^{xyz})(x+y+z)e^{xyz}, (1+xze^{xyz})(x+y+z)e^{xyz}, (1+xye^{xyz})(x+y+z)e^{xyz} \right)$$.

Step by step solution

01

Write down the given function

The given function is $$f(x, y, z) = (x + y + z)e^{x y z}$$. Our goal is to find the gradient of this function.
02

Find the partial derivative with respect to x

To find the partial derivative of the function with respect to x, we will differentiate the function with respect to x while treating y and z as constants. $$\frac{\partial f}{\partial x}=\frac{\partial (x + y + z)e^{xyz}}{\partial x}$$ To differentiate $$x$$, use product rule: $$\frac{\partial s}{\partial x}=(1)e^{xyz}+(x+y+z)yze^{xyz}=(1+yze^{xyz})(x+y+z)e^{xyz}$$
03

Find the partial derivative with respect to y

Similarly, differentiate the function with respect to y while treating x and z as constants. $$\frac{\partial f}{\partial y}=\frac{\partial (x + y + z)e^{xyz}}{\partial y}$$ To differentiate $$y$$, use product rule: $$\frac{\partial s}{\partial y}=(1)e^{xyz}+(x+y+z)xze^{xyz}=(1+xze^{xyz})(x+y+z)e^{xyz}$$
04

Find the partial derivative with respect to z

Now differentiate the function with respect to z while treating x and y as constants. $$\frac{\partial f}{\partial z}=\frac{\partial (x + y + z)e^{xyz}}{\partial z}$$ To differentiate $$z$$, use product rule: $$\frac{\partial s}{\partial z}=(1)e^{xyz}+(x+y+z)xye^{xyz}=(1+xye^{xyz})(x+y+z)e^{xyz}$$
05

Combine the partial derivatives to form the gradient

Finally, combine the partial derivatives to form the gradient of $$f(x, y, z)$$. The gradient of the function is a vector that consists of the partial derivatives as its components. $$\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right)$$ Substitute the partial derivatives found in Steps 2, 3, and 4: $$\nabla f = \left( (1+yze^{xyz})(x+y+z)e^{xyz}, (1+xze^{xyz})(x+y+z)e^{xyz}, (1+xye^{xyz})(x+y+z)e^{xyz} \right)$$ This is the gradient of the given function $$f(x, y, z) = (x + y + z)e^{xyz}$$.

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