$\text { Find } \mathrm { f } _ { \mathbf { x } } , \mathrm { f } _ { \mathbf { y } } \text {, and } \mathrm { f } _ { \mathbf { Z } }$
-$f ( x , y , z ) = x e ^ { \left( x ^ { 2 } + y ^ { 2 } + z ^ { 2 } \right) }$

Question

Quizplus · Accepted Answer

To find partial derivatives of this function, we treat the variable we want to differentiate with respect to as our variable, and consider the other variables as constants.

$\mathrm{f_x} = (1)e^{(x^2+y^2+z^2)} + (x)(2x)e^{(x^2+y^2+z^2)} = e^{(x^2+y^2+z^2)} + 2x^2e^{(x^2+y^2+z^2)} = e^{(x^2+y^2+z^2)}(1+2x^2)$

Similarly, we can obtain $\mathrm{f_y}$ and $\mathrm{f_z}$:
$$\mathrm{f_y} = e^{(x^2+y^2+z^2)}(1+2y^2)$$
$$\mathrm{f_z} = e^{(x^2+y^2+z^2)}(1+2z^2)$$

Therefore, the correct choice is B.

Find fx,fy, and fZ\text { Find } \mathrm { f } _ { \mathbf { x } } , \mathrm { f } _ { \mathbf { y } } \text {, and } \mathrm { f } _ { \mathbf { Z } } Find fx​,fy​, and fZ​

$\text { Find } \mathrm { f } _ { \mathbf { x } } , \mathrm { f } _ { \mathbf { y } } \text {, and } \mathrm { f } _ { \mathbf { Z } }$