Find the Gradient Field of the Function $$f ( x , y , z ) = \ln \left( \frac { ( x + y ) ^ { 3 } } { z ^ { 10 } } \right) + z ^ { 9 }$$

question 230

Multiple Choice

Find the gradient field of the function.
- $$f ( x , y , z ) = \ln \left( \frac { ( x + y ) ^ { 3 } } { z ^ { 10 } } \right) + z ^ { 9 }$$

• A)\({\nabla} f = \frac { 3 } { \mathrm { x } } \mathrm { i } + \frac { 3 } { \mathrm { y } } \mathrm { j } + \left( 9 \mathrm { z } ^ { 8 } + \frac { 10 } { \mathrm { z } } \right) \mathbf { k }\)
• B) \({\nabla} f = \frac { 3 } { \mathrm { x } + \mathrm { y } } \mathrm { i } + \frac { 3 } { \mathrm { x } + \mathrm { y } } \mathrm { j } + \left( 9 \mathrm { z } ^ { 8 } + \frac { 10 } { \mathrm { z } } \right) \mathrm { k }\)
• C)\({\nabla} f = \frac { 3 } { x + y } i + \frac { 3 } { x + y } j + \left( 9 z ^ { 8 } - \frac { 10 } { z } \right) k\)
• D) \({\nabla} f = \frac { 3 } { \mathrm { x } } \mathrm { i } + \frac { 3 } { \mathrm { y } } \mathrm { j } + \left( 9 \mathrm { z } ^ { 8 } - \frac { 10 } { \mathrm { z } } \right) \mathbf { k }\)

---

Definitions:

Similar Thinking Patterns

The tendency of individuals to have comparable approaches to reasoning, problem-solving, and interpreting information.

Purposeful Action

An intentional act with a clear goal or objective in mind.

Sustained Improvement

Continuous and consistent enhancement in performance, quality, or efficiency over time.

Leader Role

The responsibilities and behaviors expected from an individual who guides or directs a group towards achieving certain goals.