Find the Gradient Field of the Function $$f ( x , y , z ) = \frac { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } { x ^ { 5 } }$$

question 84

Multiple Choice

Find the gradient field of the function.
- $$f ( x , y , z ) = \frac { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } { x ^ { 5 } }$$

• A) \({\nabla} f = \frac { 7 x ^ { 2 } + 5 y ^ { 2 } + 5 z ^ { 2 } } { x ^ { 6 } } \mathbf { i } + \frac { 2 y } { x ^ { 5 } } j + \frac { 2 z } { x ^ { 5 } } \mathbf { k }\)
• B)\({\nabla} f = \frac { 5 } { x ^ { 6 } } \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) \mathbf { i } + \frac { 2 y } { x ^ { 5 } } j + \frac { 2 z } { x ^ { 5 } } \mathbf { k }\)
• C) \({\nabla} f = \frac { - 3 } { x ^ { 4 } } i + 2 y j + 2 z k\)
• D) \({\nabla} f = \frac { - 3 x ^ { 2 } - 5 y ^ { 2 } - 5 z ^ { 2 } } { x ^ { 6 } } i + \frac { 2 y } { x ^ { 5 } } j + \frac { 2 z } { x ^ { 5 } } k\)

---

Definitions:

Craft Shows

Events where artisans and makers display and sell their handmade goods, often featuring a wide range of crafts.

Variable Costs

Expenses that vary directly with the level of production or output, such as raw materials and direct labor.

Fixed Manufacturing Overhead

Fixed costs that are related to the manufacturing process, including factory rent, equipment depreciation, and salary of factory supervisors.

Financial Advantage

The benefit or edge gained in financial terms, often through investment, cost-saving strategies, or superior financial planning.