Suppose that the equation F ( x , y , z ) = 0 implicitly defines each of the three variables x , y , and z as functions of the other two: z = f ( x , y ) , y = g ( x , z ) , x = h ( y , z ) . If F is differentiable and F x , F y and F z are all nonzero, show that ∂ z ∂ x ∂ x ∂ y ∂ y ∂ z = − 1
Suppose that the equation F ( x , y , z ) = 0 implicitly defines each of the three variables x , y , and z as functions of the other two: z = f ( x , y ) , y = g ( x , z ) , x = h ( y , z ) . If F is differentiable and F x , F y and F z are all nonzero, show that ∂ z ∂ x ∂ x ∂ y ∂ y ∂ z = − 1
Solution Summary: The author explains that z=f(x,y) is a differentiable function of x&y.
Suppose that the equation
F
(
x
,
y
,
z
)
=
0
implicitly defines each of the three variables x, y, and z as functions of the other two:
z
=
f
(
x
,
y
)
,
y
=
g
(
x
,
z
)
,
x
=
h
(
y
,
z
)
.
If F is differentiable and
F
x
,
F
y
and
F
z
are all nonzero, show that
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