（＊）多變數函數的偏微分
f f x y z x y z
d f dx d y dz
x x t y y t z z t d f
dt f x
dx dt
f y
d y dt
f z
dz dt
f x y
f x
f y
f z
=
= + +
E = = =
= ∂
∂ + ∂
∂ + ∂
∂
⋅
∂ ∂ ∂
∂ ∂
∂
, ,
, ,
b g
1
b g b g b g
b g
is a function of multi - variables , , and .
if then
differential of variation arieses from the variable variation arieses from the variable variation arieses from the variable z
（＊）將單變數的微分當成多變數的微分
試著將局部的單變數函數，當成一新的變數，於是就可以將原函數視為多 變數函數，再利用上述對多變數的微分公式來微分，最後還原原來的函數 即可。
底下簡要的證明將單變數的微分當成多變數的微分的理論根據：
1) u u x v v x
F u v u v dF
dx F
u du dx
F v
dv dx
du dx
dv dx
F u v uv dF
dx F
u du dx
F v
dv
dx v du
dx u dv dx
v
u
= ∧ =
= + ⇒ = ∂
∂ + ∂
∂ = +
= ⇒ = ∂
∂ + ∂
∂ = +
= =
= =
b g b g
b g 2 2
b g 2 2
,
,
1 1
2) u u v v v x
F u v u dF
dx F
u du dx
F v
dv dx
du dx
du dv
dv dx
chain rule
= ∧ =
= ⇒ = ∂
∂ + ∂
∂ = =
= =
b g b g
b g , 2 2
1 0
Ex. f f x x x e try to differentiate f x directly by yourself
x x u x v x
f f x u v x u e d f
dx f x
dx dx
f u
du dx
f v
dv dx
e x ue x e x x e x
x
v
v v x x
= = + ⋅ ⇒
= = =
= = + ⋅
= ∂
∂ + ∂
∂ + ∂
∂
= ⋅ + ⋅ + ⋅ = + ⋅ + ⋅
b g b g b g
b g b g
b g b g b g
sin .
, sin , ,
, ,
cos cos sin
2
2 2
2
1 1 2 1 2
let then
3) d
dx f x t dt f x b x db
dx f x p x d p
dx f x t dt
p x b x
p b
, , ,
x,
b g c b g h c b g h b g
b g
z b g = ⋅ − ⋅ + z ∂ ∂
let F b p x f x t dt f t dt f x
d
dx F b p x F b
db dx
F p
d p dx
F x
dx dx f x b x db
dx f x p x d p
dx f x t dt
p
b recall
a
p b
dx d
x
x
