已知x/z=lnz/y,其中z=f(x,y)求э²z/эxэy
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x/z=lnz/y
zlnz=xy
(1+ lnz)∂z/∂x = y
∂z/∂x = y/(1+lnz) (1)
zlnz=xy
(1+ lnz)∂z/∂y = x
∂z/∂y = x/(1+lnz) (2)
from (1)
∂z/∂x = y/(1+lnz)
∂^2z/∂x∂y = [ (1+lnz)- (y/z)∂z/∂y ] /(1+lnz)^2
= [ (1+lnz)- (y/z)(x/(1+lnz)) ] /(1+lnz)^2
= ( z(1+lnz)^2- xy ) /[z(1+lnz)^3]
zlnz=xy
(1+ lnz)∂z/∂x = y
∂z/∂x = y/(1+lnz) (1)
zlnz=xy
(1+ lnz)∂z/∂y = x
∂z/∂y = x/(1+lnz) (2)
from (1)
∂z/∂x = y/(1+lnz)
∂^2z/∂x∂y = [ (1+lnz)- (y/z)∂z/∂y ] /(1+lnz)^2
= [ (1+lnz)- (y/z)(x/(1+lnz)) ] /(1+lnz)^2
= ( z(1+lnz)^2- xy ) /[z(1+lnz)^3]
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x/z=lnz/y
zlnz=xy
(1+ lnz)∂z/∂x = y
∂z/∂x = y/(1+lnz) (1)
zlnz=xy
(1+ lnz)∂z/∂y = x
∂z/∂y = x/(1+lnz) (2)
from (1)
∂z/∂x = y/(1+lnz)
∂^2z/∂x∂y = [ (1+lnz)- (y/z)∂z/∂y ] /(1+lnz)^2
= [ (1+lnz)- (y/z)(x/(1+lnz)) ] /(1+lnz)^2
= ( z(1+lnz)^2- xy ) /[z(1+lnz)^3]
zlnz=xy
(1+ lnz)∂z/∂x = y
∂z/∂x = y/(1+lnz) (1)
zlnz=xy
(1+ lnz)∂z/∂y = x
∂z/∂y = x/(1+lnz) (2)
from (1)
∂z/∂x = y/(1+lnz)
∂^2z/∂x∂y = [ (1+lnz)- (y/z)∂z/∂y ] /(1+lnz)^2
= [ (1+lnz)- (y/z)(x/(1+lnz)) ] /(1+lnz)^2
= ( z(1+lnz)^2- xy ) /[z(1+lnz)^3]
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