find $f_x(x,y,z)$, $f_y(x,y,z)$, $f_z(x,y,z)$, and $f_{yx}(x,y,z)$ for the following. $f(x,y,z)=6x^3 + 2xy…

Explanation:

Step1: Find (f_x(x,y,z))

Differentiate (f(x,y,z)=6x^{3}+2xy - 2z^{3}) with respect to (x) (treating (y) and (z) as constants). Using the power rule (\frac{d}{dx}(ax^{n})=nax^{n - 1}) and (\frac{d}{dx}(c)=0) (where (c) is a constant). (f_x(x,y,z)=\frac{\partial}{\partial x}(6x^{3}+2xy - 2z^{3})=6\times3x^{2}+2y-0 = 18x^{2}+2y)

Step2: Find (f_y(x,y,z))

Differentiate (f(x,y,z)=6x^{3}+2xy - 2z^{3}) with respect to (y) (treating (x) and (z) as constants). (f_y(x,y,z)=\frac{\partial}{\partial y}(6x^{3}+2xy - 2z^{3})=0 + 2x-0=2x)

Step3: Find (f_z(x,y,z))

Differentiate (f(x,y,z)=6x^{3}+2xy - 2z^{3}) with respect to (z) (treating (x) and (y) as constants). Using the power rule (\frac{d}{dz}(az^{n})=naz^{n - 1}). (f_z(x,y,z)=\frac{\partial}{\partial z}(6x^{3}+2xy - 2z^{3})=0+0-2\times3z^{2}=-6z^{2})

Step4: Find (f_{yx}(x,y,z))

Differentiate (f_y(x,y,z) = 2x) with respect to (x) (treating (y) and (z) as constants). (f_{yx}(x,y,z)=\frac{\partial}{\partial x}(2x)=2)

Answer:

(f_x(x,y,z)=18x^{2}+2y), (f_y(x,y,z)=2x), (f_z(x,y,z)=-6z^{2}), (f_{yx}(x,y,z)=2)