Find the gradient field of the function.
-$$f ( x , y , z ) = x ^ { 6 } y ^ { 9 } + \frac { x ^ { 6 } } { z ^ { 8 } }$$
A) ${\nabla} f = \left( 6 \mathrm { x } ^ { 5 } + 6 \mathrm { x } ^ { 5 } \right) \mathbf { i } + 9 \mathrm { y } ^ { 8 } \mathbf { j } + \frac { 8 } { \mathrm { z } ^ { 9 } } \mathbf { k }$
B) ${\nabla} f = \left( 6 \mathrm { x } ^ { 5 } + 6 \mathrm { x } ^ { 5 } \right) \mathbf { i } + 9 \mathrm { y } ^ { 8 } \mathrm { j } - \frac { 8 } { \mathrm { z } ^ { 9 } } \mathbf { k }$
C) ${\nabla} f = \left( 6 x ^ { 5 } y ^ { 9 } + \frac { 6 x ^ { 5 } } { z ^ { 8 } } \right) i + 9 x ^ { 6 } y ^ { 8 } j - \frac { 8 x ^ { 6 } } { z ^ { 9 } } k$
D)${\nabla} f = 6 x ^ { 5 } y ^ { 9 } i + 9 x ^ { 6 } y 8 - \frac { 8 x ^ { 6 } } { z ^ { 9 } } k$

Question

Find the gradient field of the function.
-$$f ( x , y , z ) = x ^ { 6 } y ^ { 9 } + \frac { x ^ { 6 } } { z ^ { 8 } }$$
A) ${\nabla} f   = \left( 6 \mathrm { x } ^ { 5 } + 6 \mathrm { x } ^ { 5 } \right) \mathbf { i } + 9 \mathrm { y } ^ { 8 } \mathbf { j } + \frac { 8 } { \mathrm { z } ^ { 9 } } \mathbf { k }$
B) ${\nabla} f = \left( 6 \mathrm { x } ^ { 5 } + 6 \mathrm { x } ^ { 5 } \right) \mathbf { i } + 9 \mathrm { y } ^ { 8 } \mathrm { j } - \frac { 8 } { \mathrm { z } ^ { 9 } } \mathbf { k }$
C) ${\nabla} f  = \left( 6 x ^ { 5 } y ^ { 9 } + \frac { 6 x ^ { 5 } } { z ^ { 8 } } \right) i + 9 x ^ { 6 } y ^ { 8 } j - \frac { 8 x ^ { 6 } } { z ^ { 9 } } k$
D)${\nabla} f = 6 x ^ { 5 } y ^ { 9 } i + 9 x ^ { 6 } y 8 - \frac { 8 x ^ { 6 } } { z ^ { 9 } } k$

Quizplus · Accepted Answer

To find the gradient field of the function, we need to take the partial derivatives of the function with respect to each variable and then combine them as follows:

$$\nabla f = \frac{\partial f}{\partial x} \mathbf{i} + \frac{\partial f}{\partial y} \mathbf{j} + \frac{\partial f}{\partial z} \mathbf{k}$$

Taking the partial derivative of f with respect to x, we get:

$$\frac{\partial f}{\partial x} = 6x^5y^9 + \frac{6x^5}{z^8}$$

Taking the partial derivative of f with respect to y, we get:

$$\frac{\partial f}{\partial y} = 9x^6y^8$$

Taking the partial derivative of f with respect to z, we get:

$$\frac{\partial f}{\partial z} = -\frac{8x^6}{z^9}$$

Combining these partial derivatives, we get the gradient field:

$$\nabla f = \left( 6 x ^ { 5 } y ^ { 9 } + \frac { 6 x ^ { 5 } } { z ^ { 8 } } \right) \mathbf{i} + 9 x ^ { 6 } y ^ { 8 } \mathbf{j} - \frac { 8 x ^ { 6 } } { z ^ { 9 } } \mathbf{k}$$

Therefore, the best choice is C.

Find the Gradient Field of the Function f(x,y,z)=x6y9+x6z8f ( x , y , z ) = x ^ { 6 } y ^ { 9 } + \frac { x ^ { 6 } } { z ^ { 8 } }f(x,y,z)=x6y9+z8x6​

Find the Gradient Field of the Function $f ( x , y , z ) = x ^ { 6 } y ^ { 9 } + \frac { x ^ { 6 } } { z ^ { 8 } }$