Find the Gradient Field of the Function $f ( x , y , z ) = \ln \left( x ^ { 4 } + y ^ { 5 } + z ^ { 6 } \right)$

question 71

Multiple Choice

Find the gradient field of the function.
- $f ( x , y , z ) = \ln \left( x ^ { 4 } + y ^ { 5 } + z ^ { 6 } \right)$

• A) \(\nabla \mathrm { f } = \frac { 4 } { x } \ln \left( y ^ { 5 } + z ^ { 6 } \right) i + \frac { 5 } { y } \ln \left( x ^ { 4 } + z ^ { 6 } \right) j + \frac { 6 } { z } \ln \left( x ^ { 4 } + y ^ { 5 } \right) \mathbf { k }\)
• B) \(\nabla f = \frac { 4 x ^ { 3 } } { x ^ { 4 } + y ^ { 5 } + z ^ { 6 } } i + \frac { 5 y ^ { 4 } } { x ^ { 4 } + y ^ { 5 } + z ^ { 6 } } j + \frac { 6 z ^ { 5 } } { x ^ { 4 } + y ^ { 5 } + z ^ { 6 } } k\)
• C) \(\nabla f = \frac { 1 } { x ^ { 4 } } i + \frac { 1 } { y ^ { 5 } } j + \frac { 1 } { z ^ { 6 } } k\)
• D) \(\nabla f = \frac { 4 } { x ^ { 4 } } i + \frac { 5 } { y ^ { 5 } } j + \frac { 6 } { z ^ { 6 } } k\)

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