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
Explanation:
-
First order derivatives:
δf(x,y,z)δx=1xyz⋅yz=1x
δf(x,y,z)δy=1xyz⋅xz=1y
δf(x,y,z)δz=1xyz⋅xy=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
-
First order derivatives:
δf(x,y,z)δx=1x2y⋅2xy=1x
δf(x,y,z)δy=1x2y⋅x2=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