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) ${\nabla} f = \frac { - 3 x ^ { 2 } - 5 y ^ { 2 } - 5 z ^ { 2 } } { x ^ { 6 } } i + \frac { 2 y } { x ^ { 5 } } j + \frac { 2 z } { x ^ { 5 } } k$

Question

Quizplus · Accepted Answer

The gradient field is found by taking the partial derivatives of $f(x, y, z)$ with respect to $x$, $y$, and $z$. The correct gradient is $\nabla f = \left(\frac{-3(x^2 + y^2 + z^2)}{x^6} + \frac{2x(x^2 + y^2 + z^2)}{x^7}\right) \mathbf{i} + \frac{2y}{x^5} \mathbf{j} + \frac{2z}{x^5} \mathbf{k}$, which simplifies to choice D.

Find the Gradient Field of the Function f(x,y,z)=x2+y2+z2x5f ( x , y , z ) = \frac { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } { x ^ { 5 } }f(x,y,z)=x5x2+y2+z2​

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