$\text { Find } \mathrm { f } _ { \mathbf { x } } , \mathrm { f } _ { \mathbf { y } } \text {, and } \mathrm { f } _ { \mathbf { Z } }$
-$f ( x , y , z ) = x ^ { 2 } y + y ^ { 2 } z + x z ^ { 2 }$
A) $f _ { x } = 2 x y + z ^ { 2 } ; f _ { y } = x ^ { 2 } + 2 y z ; f _ { z } = y ^ { 2 } + 2 x z$
B) $f _ { X } = 2 y + z ^ { 2 } ; f _ { y } = x ^ { 2 } + 2 z ; f _ { z } = y ^ { 2 } + 2 x$
C) $f _ { x } = 2 x y ; f _ { y } = x ^ { 2 } + 2 y z ; f _ { z } = y ^ { 2 } + 2 x z$
D) $f _ { x } = 2 x y + z ^ { 2 } ; f _ { y } = x ^ { 2 } + y z ; f _ { z } = y ^ { 2 } + x z$

Question

Quizplus · Accepted Answer

To find the partial derivative with respect to a variable, we differentiate the function with respect to that variable and treat the other variables as constants. Applying this rule, we get:
$$f_x=2xy+z^2$$
$$f_y=x^2+2yz$$
$$f_z=y^2+2xz$$
Therefore, option A is the correct choice.

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​

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