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) $\nabla f = \frac { 4 } { x ^ { 4 } } i + \frac { 5 } { y ^ { 5 } } j + \frac { 6 } { z ^ { 6 } } k$

Question

Quizplus · Accepted Answer

Using the chain rule, 
$$\frac{\partial f}{\partial x} = \frac{4x^3}{x^4+y^5+z^6}, \frac{\partial f}{\partial y} = \frac{5y^4}{x^4+y^5+z^6}, \text{ and } \frac{\partial f}{\partial z} = \frac{6z^5}{x^4+y^5+z^6}$$
Therefore, the gradient field of $f$ is $\nabla f = \frac{4x^3}{x^4+y^5+z^6} i + \frac{5y^4}{x^4+y^5+z^6} j + \frac{6z^5}{x^4+y^5+z^6} k.$

Find the Gradient Field of the Function f(x,y,z)=ln⁡(x4+y5+z6)f ( x , y , z ) = \ln \left( x ^ { 4 } + y ^ { 5 } + z ^ { 6 } \right)f(x,y,z)=ln(x4+y5+z6)

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