Find the gradient field of the function.
-$$\mathrm { f } ( \mathrm { x } , \mathrm { y } , \mathrm { z } ) = \mathrm { x } ^ { 5 } \mathrm { e } ^ { 7 \mathrm { x } } + \mathrm { y } ^ { 3 } \mathrm { z } ^ { 7 }$$
A) $\bar { V } = ( 5 + 7 x ) x ^ { 4 } e ^ { 7 x _ { i } } + 3 y ^ { 2 } z ^ { 7 } j + 7 y ^ { 3 } z ^ { 6 } \mathbf { k }$
B) $\nabla = ( 1 + x ) x ^ { 4 } e ^ { 7 x _ { i } } + y ^ { 2 } z ^ { 7 } j + y ^ { 3 } z ^ { 6 } k$
C) $\bar { A } = ( 5 + 7 x ) x ^ { 4 } e ^ { 7 x _ { i } } + 3 y ^ { 2 } j + 7 z ^ { 6 } \mathbf { k }$
D) $\overline { \mathrm { f } } = ( 5 + 7 x ) \mathrm { x } ^ { 4 } \mathrm { e } ^ { 7 x _ { i } } + \left( \mathrm { x } ^ { 5 } \mathrm { e } ^ { 7 x } + 3 \mathrm { y } ^ { 2 } \mathrm { z } ^ { 7 } \right) \mathrm { j } + \left( \mathrm { x } ^ { 5 } \mathrm { e } ^ { 7 x } + 7 \mathrm { y } ^ { 3 } \mathrm { z } ^ { 6 } \right) \mathbf { k }$

Question

Quizplus · Accepted Answer

The gradient field of a function $f(x, y, z)$ is found by taking the partial derivatives of $f$ with respect to $x$, $y$, and $z$, respectively. For the given function, the partial derivative with respect to $x$ is $5x^4e^{7x} + 7x^5e^{7x}$, with respect to $y$ is $3y^2z^7$, and with respect to $z$ is $7y^3z^6$, which matches option A.

Find the Gradient Field of the Function A) Vˉ=(5+7x)x4e7xi+3y2z7j+7y3z6k\bar { V } = ( 5 + 7 x ) x ^ { 4 } e ^ { 7 x _ { i } } + 3 y ^ { 2 } z ^ { 7 } j + 7 y ^ { 3 } z ^ { 6 } \mathbf { k }Vˉ=(5+7x)x4e7xi​+3y2z7j+7y3z6k

Find the Gradient Field of the Function A) $\bar { V } = ( 5 + 7 x ) x ^ { 4 } e ^ { 7 x _ { i } } + 3 y ^ { 2 } z ^ { 7 } j + 7 y ^ { 3 } z ^ { 6 } \mathbf { k }$