Problem 86 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

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 1: 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.

Step 2: 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: $$\frac{\partial s}{\partial x}=(1)e^{xyz}+(x+y+z)yze^{xyz}=(1+yze^{xyz})(x+y+z)e^{xyz}$$

Step 3: 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: $$\frac{\partial s}{\partial y}=(1)e^{xyz}+(x+y+z)xze^{xyz}=(1+xze^{xyz})(x+y+z)e^{xyz}$$

Step 4: 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: $$\frac{\partial s}{\partial z}=(1)e^{xyz}+(x+y+z)xye^{xyz}=(1+xye^{xyz})(x+y+z)e^{xyz}$$

Step 5: 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 $$f(x, y, z) = (x + y + z)e^{xyz}$$.