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An alternative proof for the chain rule for multivariable functions
Raymond Jensen
Northern State University
Chain rule: let f be differentiable wrt. x, y, and x, y differentiable wrt. t.
df f dx f dy
dt x dt y dt
x x t
y y t
,f f x y
Proof
0
0
0 00 0
0
, ,, lim
g t g t
t t
f x t y t f x t y tdfx t y t
dt t t
,f x t y t g t
0
00
0
limt t
g t g tdgt
dt t t
Proof: begin as with the product rule
0
0 0
0 0 0
0 0 0 0
0
0 0 0 0
0 0
,
, , , ,lim
, , , ,lim lim
t t
t t t t
df dfx t y t
dt dt
f x t y t f x t y t f x t y t f x t y t
t t
f x t y t f x t y t f x t y t f x t y t
t t t t
r
Proof: continue as with the single-variable chain rule
0
0
0 0
0 0
0 00
0 0
0 0 0 0
0 0
0 0
0 0
0 0 0 0
0 0
, ,lim
, ,lim
, ,lim lim
, ,lim lim
t t
t t
t t
t t
f x t y t f x t y t x t x tdf
dt x t x t t t
f x t y t f x t y t y t y t
y t y t t t
f x y f x y x t x t
x x t t
f x y f x y y t y t
y y t t
r r
r r
r
Proof: continue as with the single-variable chain rule
0 0
0 0
0 00
0 0
" "
0 0 0 0
0 0
, ,lim lim
, ,lim lim
t t
f dxx dt
t t
f dyy dt
f x y f x y x t x tdf
dt x x t t
f x y f x y y t y t
y y t t
r r
r r
r
Definition: f is [new] differentiable at r0 if
0
0
00
0
00
0
, ,lim
and
, ,lim
f x y f x y f
x x x
f x y f x y f
y y y
r r
r r
r
r
Definition: f is [old] differentiable at r0 if
0 0 0
0 0 1 2
1 1 2 2
0 0 0
where , 0 as
, ,
f f f
f fx y x y
x y
x x x y y y
r r r r
r r
r r r 0
r r r
Old differentiability chain rule
0 0 0 1 2
0 0 0
f f x f y x y
t x t y t t t
df f dx f dy
dt x dt y dt
r r r
r r r
Old differentiability continuity
• Leithold thm. 12.4.3
New differentiability continuity
0 0
0 0
0 0
0
0 0 0 0 0
0 0 0 0
00
00
0 0 0
0
lim lim , , , ,
lim , , lim , ,
, ,lim lim
, ,lim
fy
fx
f f f x y f x y f x y f x y
f x y f x y f x y f x y
f x y f x yy y
y y
f x y f x y
x x
r r r r
r r r r
r r r r
r r
r r
00
0
lim
0
x x
r r
If partials of f exist near r0 and are continuous at r0 then f is old differentiable at r0.
• Stewart thm. 11.4.8, Leithold thm. 12.4.4
If partials of f exist near r0 and are continuous at r0 then f is new differentiable at r0.
0 0 0
0 0
0
0
0 0 0 0 00
0 0 0
0 0 0 0
0
0 0
0
, , , , , ,lim lim lim
, , , ,lim
lim
g x g x h x h x
f x y f x y f x y f x y f x y f x yf
x x x x x x x
f x y f x y f x y f x y
x x
g x g x h x h x
x x
r r r r r r
r r
r r
r
MVT
If partials exist near r0 and are continuous at r0 then f is new differentiable at r0.
0 0
0
0 0
00
0
0
0 0
, ,lim lim
lim
lim , lim ,
0
f x y f x y g u x h v xf
x x x x
g u h v
f fu y v y
x xf f
x x
r r r r
r r
r r r r
r
r r
Old differentiability new differentiability
0
0
0
0 0 0
00
0
0 0 0 0
0
0 0 0 0 0 0
0
0 0 0 0 0 0 0 0
0 0 0
, ,lim
, , , ,lim
, , , , 2 ,lim
, , , , , ,lim lim lim
lim
fy
f x y f x y f
y y y
f x y f x y f x y f x y
y y
f x y f x y f x y f x y f x y
y y
f x y f x y f x y f x y f x y f x y
y y y y y y
r r
r r
r r
r r r r r r
r
r
0 0 0
0 0 0 0 0 00
0 0 0
, , , ,lim lim
fx
f x y f x y f x y f x y x x f
y y x x y y y
r r r r rr
Old differentiability new differentiability
0 0
0 0 00 0
0 0
sin /
, ,lim lim
h b a
f x y f x y x xf f
y y x y y y
r r r rr r
Old differentiability new differentiability
0
0 00 0
, ,lim
sin
f x y f x y b f f
h a x y
r rr r
Old differentiability new differentiability
0 0 0 00 00
, ,1lim
sin h
D f
f x bh y ah f x y b f f
h a x y
u
r r
,b au
Old differentiability new differentiability
0 0 0
1 b f fD f
a a x y
u r r r
,b au
Thm: If f is old differentiable at r0 then Duf(r0) exists and:
• Stewart thm. 11.6.3, Leithold thm. 12.6.2• u = (b, a)
0 0 0
0
f fD f b a
x y
f
u r r r
r u
Old differentiability new differentiability
0
00
0
0 0 0 0
, ,lim
1
0.
f x y f x y f
y y y
f f b f fb a
a x y a x y
r rr
r r r r
Conclusion
• Old differentiability new differentiability chain rule.
Question:
• Old differentiability new differentiability?
References
• Stewart, Essential Calculus (Thompson, 2007)• Leithold, The Calculus 7th ed. (Harper, 1996)
