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)=4x^5 + 3xy…

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 $f(x,y,z)=4x^5 + 3xy - 3z^5$ $f_x(x,y,z)=$ $f_y(x,y,z)=$ $f_z(x,y,z)=$ $f_{yx}(x,y,z)=$

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 $f(x,y,z)=4x^5 + 3xy - 3z^5$ $f_x(x,y,z)=$ $f_y(x,y,z)=$ $f_z(x,y,z)=$ $f_{yx}(x,y,z)=$

Answer

Explanation:

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

Differentiate (f(x,y,z) = 4x^5+3xy - 3z^5) with respect to (x). Using the power rule (\frac{d}{dx}(x^n)=nx^{n - 1}) and treating (y) and (z) as constants. (f_x(x,y,z)=\frac{\partial}{\partial x}(4x^5)+\frac{\partial}{\partial x}(3xy)-\frac{\partial}{\partial x}(3z^5)) (f_x(x,y,z)=4\times5x^{4}+3y-0 = 20x^{4}+3y)

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

Differentiate (f(x,y,z) = 4x^5+3xy - 3z^5) with respect to (y). Using the power rule and treating (x) and (z) as constants. (f_y(x,y,z)=\frac{\partial}{\partial y}(4x^5)+\frac{\partial}{\partial y}(3xy)-\frac{\partial}{\partial y}(3z^5)) (f_y(x,y,z)=0 + 3x-0=3x)

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

Differentiate (f(x,y,z) = 4x^5+3xy - 3z^5) with respect to (z). Using the power rule and treating (x) and (y) as constants. (f_z(x,y,z)=\frac{\partial}{\partial z}(4x^5)+\frac{\partial}{\partial z}(3xy)-\frac{\partial}{\partial z}(3z^5)) (f_z(x,y,z)=0+0-3\times5z^{4}=- 15z^{4})

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

Differentiate (f_y(x,y,z) = 3x) with respect to (x). Treating (y) and (z) as constants. (f_{yx}(x,y,z)=\frac{\partial}{\partial x}(3x)=3)

Answer:

(f_x(x,y,z)=20x^{4}+3y), (f_y(x,y,z) = 3x), (f_z(x,y,z)=-15z^{4}), (f_{yx}(x,y,z)=3)