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:
    δf(x,y,z)δx=1xyzyz=1x
    δf(x,y,z)δy=1xyzxz=1y
    δf(x,y,z)δz=1xyzxy=1z

  • Second order derivatives:
    δf(x,y,z)2δ2x=1x2
    δf(x,y,z)2δ2y=1y2
    δf(x,y,z)2δ2z=1z2

  • Third order derivatives:
    δf(x,y,z)3δ3x=2x3
    δf(x,y,z)3δ3y=2y3
    δf(x,y,z)3δ3z=2z3

BUT
If
your function is actually f(x,y,z)=ln(xyx)=ln(x2y), then...

  • First order derivatives:
    δf(x,y,z)δx=1x2y2xy=1x
    δf(x,y,z)δy=1x2yx2=1y
    δf(x,y,z)δz=0

  • Second order derivatives:
    δf(x,y,z)2δ2x=1x2
    δf(x,y,z)2δ2y=1y2
    δf(x,y,z)2δ2z=0

  • Third order derivatives:
    δf(x,y,z)3δ3x=2x3
    δf(x,y,z)3δ3y=2y3
    δf(x,y,z)3δ3z=0