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.\n( f(x,y,z)=3x^{3}+2xy…

Explanation:

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

Differentiate (f(x,y,z) = 3x^{3}+2xy - 4z^{4}) with respect to (x) (treating (y) and (z) as constants). Using the power rule (\frac{d}{dx}(x^{n})=nx^{n - 1}) and (\frac{d}{dx}(ax)=a) (where (a) is a constant), we have: (f_x(x,y,z)=\frac{\partial}{\partial x}(3x^{3}+2xy - 4z^{4})=3\times3x^{2}+2y-0 = 9x^{2}+2y)

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

Differentiate (f(x,y,z) = 3x^{3}+2xy - 4z^{4}) with respect to (y) (treating (x) and (z) as constants). (f_y(x,y,z)=\frac{\partial}{\partial y}(3x^{3}+2xy - 4z^{4})=0 + 2x-0=2x)

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

Differentiate (f(x,y,z) = 3x^{3}+2xy - 4z^{4}) with respect to (z) (treating (x) and (y) as constants). Using the power rule (\frac{d}{dz}(z^{n})=nz^{n - 1}), we get (f_z(x,y,z)=\frac{\partial}{\partial z}(3x^{3}+2xy - 4z^{4})=0+0-4\times4z^{3}=-16z^{3})

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

First, we know (f_y(x,y,z) = 2x). Then differentiate (f_y(x,y,z)) 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)=9x^{2}+2y), (f_y(x,y,z)=2x), (f_z(x,y,z)=-16z^{3}), (f_{yx}(x,y,z) = 2)