Question

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. $f(x,y,z)=6x^3 + 2xy - 2z^3$ $f_x(x,y,z)=$ $f_y(x,y,z)=$ $f_z(x,y,z)=$ $f_{yx}(x,y,z)=$

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:

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