Find the gradient vector field of $f$. Select the correct answer.
$$f ( x , y , z ) = \sqrt { x ^ { 2 } + 4 y ^ { 2 } + 6 z ^ { 2 } }$$
A) $\frac { y } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \mathbf { i } + \frac { x } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \mathbf { j } + \frac { z } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \mathbf { k }$
B) $\frac { 1 } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \mathbf { i } + \frac { 1 } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \mathbf { j } + \frac { z } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \mathbf { k }$
C)
$\frac { 2 x } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \mathbf { i } + \frac { 2 y } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \mathbf { j } + \frac { 2 z } { \sqrt { x ^ { 2 } + y ^ { 2 } + z ^ { 2 } } } \mathbf { k }$
D)
$\frac { x } { \sqrt { x ^ { 2 } + 4 y ^ { 2 } + 6 z ^ { 2 } } } \mathbf { i } + \frac { 4 y } { \sqrt { x ^ { 2 } + 4 y ^ { 2 } + 6 z ^ { 2 } } } \mathbf { j } + \frac { 6 z } { \sqrt { x ^ { 2 } + 4 y ^ { 2 } + 6 z ^ { 2 } } } \mathbf { k }$
E) None of these

Question

Quizplus · Accepted Answer

By definition, the gradient vector of a scalar function is the vector composed of its partial derivatives with respect to each variable. Therefore, we have:
$$

abla f = \frac{\partial f}{\partial x} \mathbf{i} + \frac{\partial f}{\partial y} \mathbf{j} + \frac{\partial f}{\partial z} \mathbf{k}
$$
Taking the partial derivative with respect to $x$, we get:
$$
\frac{\partial f}{\partial x} = \frac{x}{\sqrt{x^2+4y^2+6z^2}}
$$
Similarly, for $y$ and $z$, we have:
$$
\frac{\partial f}{\partial y} = \frac{4y}{\sqrt{x^2+4y^2+6z^2}} \quad 	ext{and} \quad \frac{\partial f}{\partial z} = \frac{6z}{\sqrt{x^2+4y^2+6z^2}}
$$
Thus, the gradient vector is:
$$

abla f = \frac{x}{\sqrt{x^2+4y^2+6z^2}} \mathbf{i} + \frac{4y}{\sqrt{x^2+4y^2+6z^2}} \mathbf{j} + \frac{6z}{\sqrt{x^2+4y^2+6z^2}} \mathbf{k}
$$

Find the Gradient Vector Field Of fff Select the Correct Answer

Find the Gradient Vector Field Of $f$ Select the Correct Answer