What are the first, second, and third order partial derivatives of #f(x,y,z)=ln(xyx)#?

1 Answer
Dec 30, 2015

Supposing it is #f(x,y,z)=ln(xyz)#...

Explanation:

  • First order derivatives:
    #(deltaf(x,y,z))/(deltax)=1/(xyz)*yz=1/x#
    #(deltaf(x,y,z))/(deltay)=1/(xyz)*xz=1/y#
    #(deltaf(x,y,z))/(deltaz)=1/(xyz)*xy=1/z#

  • Second order derivatives:
    #(deltaf(x,y,z)^2)/(delta^2x)=-1/x^2#
    #(deltaf(x,y,z)^2)/(delta^2y)=-1/y^2#
    #(deltaf(x,y,z)^2)/(delta^2z)=-1/z^2#

  • Third order derivatives:
    #(deltaf(x,y,z)^3)/(delta^3x)=2/x^3#
    #(deltaf(x,y,z)^3)/(delta^3y)=2/y^3#
    #(deltaf(x,y,z)^3)/(delta^3z)=2/z^3#

BUT
If
your function is actually #f(x,y,z)=ln(xyx)=ln(x^2y)#, then...

  • First order derivatives:
    #(deltaf(x,y,z))/(deltax)=1/(x^2y)*2xy=1/x#
    #(deltaf(x,y,z))/(deltay)=1/(x^2y)*x^2=1/y#
    #(deltaf(x,y,z))/(deltaz)=0#

  • Second order derivatives:
    #(deltaf(x,y,z)^2)/(delta^2x)=-1/x^2#
    #(deltaf(x,y,z)^2)/(delta^2y)=-1/y^2#
    #(deltaf(x,y,z)^2)/(delta^2z)=0#

  • Third order derivatives:
    #(deltaf(x,y,z)^3)/(delta^3x)=2/x^3#
    #(deltaf(x,y,z)^3)/(delta^3y)=2/y^3#
    #(deltaf(x,y,z)^3)/(delta^3z)=0#