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Chapter 13: Problem 85

Use the gradient rules of Exercise 81 to find the gradient of the following functions. $$f(x, y, z)=\sqrt{25-x^{2}-y^{2}-z^{2}}$$

Short Answer

Expert verified
Question: Calculate the gradient of the function $$f(x, y, z)=\sqrt{25-x^{2}-y^{2}-z^{2}}$$. Answer: The gradient of the function f(x, y, z) is $$\nabla f = \left\langle \frac{-x}{\sqrt{25-x^2-y^2-z^2}}, \frac{-y}{\sqrt{25-x^2-y^2-z^2}}, \frac{-z}{\sqrt{25-x^2-y^2-z^2}} \right\rangle$$.

Step by step solution

01

Calculate the partial derivative with respect to x.

To find the partial derivative of f(x, y, z) with respect to x, we will differentiate the function with respect to x while treating y and z as constants. This is represented as: $$\frac{\partial f}{\partial x}$$ Using the chain rule, we have: $$\frac{\partial f}{\partial x} = \frac{-x}{\sqrt{25-x^2-y^2-z^2}}$$
02

Calculate the partial derivative with respect to y.

Next, we will find the partial derivative of f(x, y, z) with respect to y. We differentiate the function with respect to y while treating x and z as constants. This is represented as: $$\frac{\partial f}{\partial y}$$ Using the chain rule, we obtain: $$\frac{\partial f}{\partial y} = \frac{-y}{\sqrt{25-x^2-y^2-z^2}}$$
03

Calculate the partial derivative with respect to z.

Lastly, we will find the partial derivative of f(x, y, z) with respect to z. Differentiate the function with respect to z while treating x and y as constants. This is represented as: $$\frac{\partial f}{\partial z}$$ Using the chain rule, we have: $$\frac{\partial f}{\partial z} = \frac{-z}{\sqrt{25-x^2-y^2-z^2}}$$
04

Determine the gradient vector.

The gradient vector is formed by combining the partial derivatives that we calculated in the previous steps. The gradient of the function f(x, y, z) is: $$\nabla f = \left\langle \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}, \frac{\partial f}{\partial z} \right\rangle$$ Substitute the partial derivatives from Steps 1, 2, and 3: $$\nabla f = \left\langle \frac{-x}{\sqrt{25-x^2-y^2-z^2}}, \frac{-y}{\sqrt{25-x^2-y^2-z^2}}, \frac{-z}{\sqrt{25-x^2-y^2-z^2}} \right\rangle$$

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