Question

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

Answer 1

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`