Answer:
[tex]f_x(x,y,z)=4\sin (y-z)[/tex]
[tex]f_x(x,y,z)=4x\cos (y-z)[/tex]
[tex]f_z(x,y,z)=-4x\cos (y-z)[/tex]
Step-by-step explanation:
The given function is
[tex]f(x,y,z)=4x\sin (y-z)[/tex]
We need to find first partial derivatives of the function.
Differentiate partially w.r.t. x and y, z are constants.
[tex]f_x(x,y,z)=4(1)\sin (y-z)[/tex]
[tex]f_x(x,y,z)=4\sin (y-z)[/tex]
Differentiate partially w.r.t. y and x, z are constants.
[tex]f_y(x,y,z)=4x\cos (y-z)\dfrac{\partial}{\partial y}(y-z)[/tex]
[tex]f_y(x,y,z)=4x\cos (y-z)[/tex]
Differentiate partially w.r.t. z and x, y are constants.
[tex]f_z(x,y,z)=4x\cos (y-z)\dfrac{\partial}{\partial z}(y-z)[/tex]
[tex]f_z(x,y,z)=4x\cos (y-z)(-1)[/tex]
[tex]f_z(x,y,z)=-4x\cos (y-z)[/tex]
Therefore, the first partial derivatives of the function are [tex]f_x(x,y,z)=4\sin (y-z), f_x(x,y,z)=4x\cos (y-z)\text{ and }f_z(x,y,z)=-4x\cos (y-z)[/tex].
