What are the first, second, and third order partial derivatives of #f(x,y,z)=ln(xyx)#?
1 Answer
Supposing it is
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
-
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#
