Calculo 2 De dos variables_9na Edición - Ron Larson

ApénDiCE A Demostración de teoremas seleccionados A-3
Del mismo modo, desde B hasta C, x es fija, y cambia, y existe un valor y 1 entre y y y + Dy
tal que
Combinando estos dos resultados, escribir
Definiendo e 1 y e 2 como
Between and is fixed and changes. So, by the Mean Value Theorem, there is
a value between and such that
Similarly, between and is fixed and changes, and there is a value between
and
such that
By combining these two results, you can write
If you define and as and
it follows that
By the continuity of and and the fact that and
it follows that and as and Therefore, by definition, is
differentiable.
Because and are differentiable functions of you know that both
and
approach zero as approaches zero. Moreover, because is a differentiable
function of and
you know that
where both and as So, for
from which it follows that
w
x
dx
dt
w
y
dy
dt .
dw
dt
lím
t→0
w
t
w
x
dx
dt
w
y
dy
dt
0 dx
dt
0 dy
dt
w
t
w
x
x
t
w
y
y
t
1
x
t
2
y
t
t 0
x, y → 0, 0 .
2 → 0
1
w w x x w y y 1 x 2 y,
y,
x
f
t
y
x
t,
h
g
PROOF
f
y → 0.
x → 0
2 → 0
1→ 0
y y,
y y 1
x x 1 x x
f y
f x f x x, y x f y x, y y 1 x 2 y.
z z 1 z 2 1 f x x, y x 2 f y x, y y
2 f y x x, y 1 f y x, y ,
1 f x x 1 , y f x x, y
2
1
z z 1 z 2 f x x 1 , y x f y x x, y 1 y.
z 2 f x x, y y f x x, y f y x x, y 1 y.
y
y
y
y 1
y
x
C,
B
z 1 f x x, y f x, y f x x 1 , y x.
x
x
x
x 1
x
y
B,
A
z 1 z 2 .
THEOREM 13.6 CHAIN RULE: ONE INDEPENDENT VARIABLE (PAGE 925)
Let where is a differentiable function of and If and
where and are differentiable functions of then is a differentiable
function of
and
dw
dt
w
x
dx
dt
w
y
dy
dt .
t,
w
t,
h
g
y h t ,
x
g t
y.
x
f
w f x, y ,
y
Between and is fixed and changes. So, by the Mean Value Theorem, there is
a value between and such that
Similarly, between and is fixed and changes, and there is a value between
and
such that
By combining these two results, you can write
If you define and as and
it follows that
By the continuity of and and the fact that and
it follows that and as and Therefore, by definition, is
differentiable.
Because and are differentiable functions of you know that both
and
approach zero as approaches zero. Moreover, because is a differentiable
function of and
you know that
where both and as So, for
from which it follows that
w
x
dx
dt
w
y
dy
dt .
dw
dt
lím
t→0
w
t
w
x
dx
dt
w
y
dy
dt
0 dx
dt
0 dy
dt
w
t
w
x
x
t
w
y
y
t
1
x
t
2
y
t
t 0
x, y → 0, 0 .
2 → 0
1
w w x x w y y 1 x 2 y,
y,
x
f
t
y
x
t,
h
g
PROOF
f
y → 0.
x → 0
2 → 0
1→ 0
y y,
y y 1
x x 1 x x
f y
f x f x x, y x f y x, y y 1 x 2 y.
z z 1 z 2 1 f x x, y x 2 f y x, y y
2 f y x x, y 1 f y x, y ,
1 f x x 1 , y f x x, y
2
1
z z 1 z 2 f x x 1 , y x f y x x, y 1 y.
z 2 f x x, y y f x x, y f y x x, y 1 y.
y
y
y
y 1
y
x
C,
B
z 1 f x x, y f x, y f x x 1 , y x.
x
x
x
x 1
x
y
B,
A
z 1 z 2 .
THEOREM 13.6 CHAIN RULE: ONE INDEPENDENT VARIABLE (PAGE 925)
Let where is a differentiable function of and If and
where and are differentiable functions of then is a differentiable
function of
and
dw
dt
w
x
dx
dt
w
y
dy
dt .
t,
w
t,
h
g
y h t ,
x
g t
y.
x
f
w f x, y ,
se deduce
que
por medio de la continuidad de f x y f y y del hecho de que
Between and is fixed and changes. So, by the Mean Value Theorem, there is
a value between and such that
Similarly, between and is fixed and changes, and there is a value between
and
such that
By combining these two results, you can write
If you define and as and
it follows that
By the continuity of and and the fact that and
it follows that and as and Therefore, by definition, is
differentiable.
Because and are differentiable functions of you know that both
and
approach zero as approaches zero. Moreover, because is a differentiable
function of and
you know that
where both and as So, for
from which it follows that
w
x
dx
dt
w
y
dy
dt .
dw
dt
lím
t→0
w
t
w
x
dx
dt
w
y
dy
dt
0 dx
dt
0 dy
dt
w
t
w
x
x
t
w
y
y
t
1
x
t
2
y
t
t 0
x, y → 0, 0 .
2 → 0
1
w w x x w y y 1 x 2 y,
y,
x
f
t
y
x
t,
h
g
PROOF
f
y → 0.
x → 0
2 → 0
1→ 0
y y,
y y 1
x x 1 x x
f y
f x f x x, y x f y x, y y 1 x 2 y.
z z 1 z 2 1 f x x, y x 2 f y x, y y
2 f y x x, y 1 f y x, y ,
1 f x x 1 , y f x x, y
2
1
z z 1 z 2 f x x 1 , y x f y x x, y 1 y.
z 2 f x x, y y f x x, y f y x x, y 1 y.
y
y
y
y 1
y
x
C,
B
z 1 f x x, y f x, y f x x 1 , y x.
x
x
x
x 1
x
y
B,
A
z 1 z 2 .
THEOREM 13.6 CHAIN RULE: ONE INDEPENDENT VARIABLE (PAGE 925)
Let where is a differentiable function of and If and
where and are differentiable functions of then is a differentiable
function of
and
dw
dt
w
x
dx
dt
w
y
dy
dt .
t,
w
t,
h
g
y h t ,
x
g t
y.
x
f
w f x, y ,
y
Between and is fixed and changes. So, by the Mean Value Theorem, there is
a value between and such that
Similarly, between and is fixed and changes, and there is a value between
and
such that
By combining these two results, you can write
If you define and as and
it follows that
By the continuity of and and the fact that and
it follows that and as and Therefore, by definition, is
differentiable.
Because and are differentiable functions of you know that both
and
approach zero as approaches zero. Moreover, because is a differentiable
function of and
you know that
where both and as So, for
from which it follows that
w
x
dx
dt
w
y
dy
dt .
dw
dt
lím
t→0
w
t
w
x
dx
dt
w
y
dy
dt
0 dx
dt
0 dy
dt
w
t
w
x
x
t
w
y
y
t
1
x
t
2
y
t
t 0
x, y → 0, 0 .
2 → 0
1
w w x x w y y 1 x 2 y,
y,
x
f
t
y
x
t,
h
g
PROOF
f
y → 0.
x → 0
2 → 0
1→ 0
y y,
y y 1
x x 1 x x
f y
f x f x x, y x f y x, y y 1 x 2 y.
z z 1 z 2 1 f x x, y x 2 f y x, y y
2 f y x x, y 1 f y x, y ,
1 f x x 1 , y f x x, y
2
1
z z 1 z 2 f x x 1 , y x f y x x, y 1 y.
z 2 f x x, y y f x x, y f y x x, y 1 y.
y
y
y
y 1
y
x
C,
B
z 1 f x x, y f x, y f x x 1 , y x.
x
x
x
x 1
x
y
B,
A
z 1 z 2 .
THEOREM 13.6 CHAIN RULE: ONE INDEPENDENT VARIABLE (PAGE 925)
Let where is a differentiable function of and If and
where and are differentiable functions of then is a differentiable
function of
and
dw
dt
w
x
dx
dt
w
y
dy
dt .
t,
w
t,
h
g
y h t ,
x
g t
y.
x
f
w f x, y ,
and
changes. So, by the Mean Value Theorem, there is
such that
is fixed and changes, and there is a value
between
lts, you can write
and
nd the fact that
and
as and Therefore, by definition, is
differentiable functions of you know that both
and
proaches zero. Moreover, because
is a differentiable
that
So, for
w
x
dx
dt
w
y
dy
dt .
w
y
dy
dt
0 dx
dt
0 dy
dt
1
x
t
2
y
t
t 0
x, y → 0, 0 .
w w x x w y y 1 x 2 y,
f
x
t,
f
y → 0.
x → 0
0
y y,
y y 1
x x 1 x x
x, y x f y x, y y 1 x 2 y.
f x x, y x 2 f y x, y y
2 f y x x, y 1 f y x, y ,
f x x 1 , y
f x x, y
1 , y x f y x x, y 1 y.
y f x x, y f y x x, y 1 y.
y 1
y
x
x, y f x x 1 , y x.
x
x
x, y f x x, y y f x x, y
ULE: ONE INDEPENDENT VARIABLE (PAGE 925)
s a differentiable function of and If and
re differentiable functions of then is a differeny
t .
w
t,
x
g t
y.
x
se deduce que
Between and is fixed and changes. So, by the Mean Value Theorem, there is
a value between and such that
Similarly, between and is fixed and changes, and there is a value between
and
such that
By combining these two results, you can write
If you define and as and
it follows that
By the continuity of and and the fact that and
it follows that and as and Therefore, by definition, is
differentiable.
Because and are differentiable functions of you know that both
and
approach zero as approaches zero. Moreover, because is a differentiable
function of and
you know that
where both and as So, for
from which it follows that
w
x
dx
dt
w
y
dy
dt .
dw
dt
lím
t→0
w
t
w
x
dx
dt
w
y
dy
dt
0 dx
dt
0 dy
dt
w
t
w
x
x
t
w
y
y
t
1
x
t
2
y
t
t 0
x, y → 0, 0 .
2 → 0
1
w w x x w y y 1 x 2 y,
y,
x
f
t
y
x
t,
h
g
PROOF
f
y → 0.
x → 0
2 → 0
1→ 0
y y,
y y 1
x x 1 x x
f y
f x f x x, y x f y x, y y 1 x 2 y.
z z 1 z 2 1 f x x, y x 2 f y x, y y
2 f y x x, y 1 f y x, y ,
1 f x x 1 , y f x x, y
2
1
z z 1 z 2 f x x 1 , y x f y x x, y 1 y.
z 2 f x x, y y f x x, y f y x x, y 1 y.
y
y
y
y 1
y
x
C,
B
z 1 f x x, y f x, y f x x 1 , y x.
x
x
x
x 1
x
y
B,
A
z 1 z 2 .
THEOREM 13.6 CHAIN RULE: ONE INDEPENDENT VARIABLE (PAGE 925)
Let where is a differentiable function of and If and
where and are differentiable functions of then is a differentiable
function of
and
dw
dt
w
x
dx
dt
w
y
dy
dt .
t,
w
t,
h
g
y h t ,
x
g t
y.
x
f
w f x, y ,
y
Between and is fixed and changes. So, by the Mean Value Theorem, there is
a value between and such that
Similarly, between and is fixed and changes, and there is a value between
and
such that
By combining these two results, you can write
If you define and as and
it follows that
By the continuity of and and the fact that and
it follows that and as and Therefore, by definition, is
differentiable.
Because and are differentiable functions of you know that both
and
approach zero as approaches zero. Moreover, because is a differentiable
function of and
you know that
where both and as So, for
from which it follows that
w
x
dx
dt
w
y
dy
dt .
dw
dt
lím
t→0
w
t
w
x
dx
dt
w
y
dy
dt
0 dx
dt
0 dy
dt
w
t
w
x
x
t
w
y
y
t
1
x
t
2
y
t
t 0
x, y → 0, 0 .
2 → 0
1
w w x x w y y 1 x 2 y,
y,
x
f
t
y
x
t,
h
g
PROOF
f
y → 0.
x → 0
2 → 0
1→ 0
y y,
y y 1
x x 1 x x
f y
f x f x x, y x f y x, y y 1 x 2 y.
z z 1 z 2 1 f x x, y x 2 f y x, y y
2 f y x x, y 1 f y x, y ,
1 f x x 1 , y f x x, y
2
1
z z 1 z 2 f x x 1 , y x f y x x, y 1 y.
z 2 f x x, y y f x x, y f y x x, y 1 y.
y
y
y
y 1
y
x
C,
B
z 1 f x x, y f x, y f x x 1 , y x.
x
x
x
x 1
x
y
B,
A
z 1 z 2 .
THEOREM 13.6 CHAIN RULE: ONE INDEPENDENT VARIABLE (PAGE 925)
Let where is a differentiable function of and If and
where and are differentiable functions of then is a differentiable
function of
and
dw
dt
w
x
dx
dt
w
y
dy
dt .
t,
w
t,
h
g
y h t ,
x
g t
y.
x
f
w f x, y ,
cuando lo hacen
Between and is fixed and changes. So, by the Mean Value Theorem, the
a value between and such that
Similarly, between and is fixed and changes, and there is a value betw
and
such that
By combining these two results, you can write
If you define and as and
it follows that
By the continuity of and and the fact that and
it follows that and as and Therefore, by definition
differentiable.
Because and are differentiable functions of you know that both
approach zero as approaches zero. Moreover, because is a different
function of and
you know that
where both and as So, for
from which it follows that
w
x
dx
dt
w
y
dy
dt .
dw
dt
lím
t→0
w
t
w
x
dx
dt
w
y
dy
dt
0 dx
dt
0 dy
dt
w
t
w
x
x
t
w
y
y
t
1
x
t
2
y
t
t 0
x, y → 0, 0 .
2 → 0
1
w w x x w y y 1 x
y,
x
f
t
y
x
t,
h
g
PROOF
y → 0.
x → 0
2 → 0
1→ 0
y
y y 1
x x 1 x x
f y
f x f x x, y x f y x, y y 1 x 2 y.
z z 1 z 2 1 f x x, y x 2 f y x, y y
2 f y x x, y 1 f y
1 f x x 1 , y f x x, y
2
1
z z 1 z 2 f x x 1 , y x f y x x, y 1 y.
z 2 f x x, y y f x x, y f y x x, y 1 y.
y
y
y
y 1
y
x
C,
B
z 1 f x x, y f x, y f x x 1 , y x.
x
x
x
x 1
x
y
B,
A
z 1 z 2 .
THEOREM 13.6 CHAIN RULE: ONE INDEPENDENT VARIABLE (PAGE 925)
Let where is a differentiable function of and If an
where and are differentiable functions of then is a differentiable
function of
and
dw
dt
w
x
dx
dt
w
y
dy
dt .
t,
w
t,
h
g
y h t ,
x
g t
y.
x
f
w f x, y ,
y
Between and is fixed and changes. So, by the Mean Value Theorem, there
a value between and such that
Similarly, between and is fixed and changes, and there is a value betwe
and
such that
By combining these two results, you can write
If you define and as and
it follows that
By the continuity of and and the fact that and
it follows that and as and Therefore, by definition,
differentiable.
Because and are differentiable functions of you know that both
a
approach zero as approaches zero. Moreover, because is a differentia
function of and
you know that
where both and as So, for
from which it follows that
w
x
dx
dt
w
y
dy
dt .
dw
dt
lím
t→0
w
t
w
x
dx
dt
w
y
dy
dt
0 dx
dt
0 dy
dt
w
t
w
x
x
t
w
y
y
t
1
x
t
2
y
t
t 0
x, y → 0, 0 .
2 → 0
1
w w x x w y y 1 x 2
y,
x
f
t
y
x
t,
h
g
PROOF
y → 0.
x → 0
2 → 0
1→ 0
y
y y 1
x x 1 x x
f y
f x f x x, y x f y x, y y 1 x 2 y.
z z 1 z 2 1 f x x, y x 2 f y x, y y
2 f y x x, y 1 f y x,
1 f x x 1 , y f x x, y
2
1
z z 1 z 2 f x x 1 , y x f y x x, y 1 y.
z 2 f x x, y y f x x, y f y x x, y 1 y.
y
y
y
y 1
y
x
C,
B
z 1 f x x, y f x, y f x x 1 , y x.
x
x
x
x 1
x
y
B,
A
z 1 z 2 .
THEOREM 13.6 CHAIN RULE: ONE INDEPENDENT VARIABLE (PAGE 925)
Let where is a differentiable function of and If and
where and are differentiable functions of then is a differentiable
function of
and
dw
dt
w
x
dx
dt
w
y
dy
dt .
t,
w
t,
h
g
y h t ,
x
g t
y.
x
f
w f x, y ,
De tal modo,
por definición, f es derivable.
TEOREMA 13.6 REglA dE lA CAdEnA: unA vARiAblE indEpEndiEnTE (páginA 925)
Sea w = f (x, y) donde f es una función diferenciable de x y y. Si x = g (t) y y = h (t),
donde g y h son funciones diferenciables de t, entonces w es una función diferenciable
de t, y
DEMOSTRACIÓN puesto que g y h son funciones diferenciables de t, se sabe que Dx y Dy
tienden a cero a medida que lo hace Dt. Además, como ƒ es una función derivable de x y y,
se sabe que
Between and is fixed and changes. So, by the Mean Value Theorem, there is
a value between and such that
Similarly, between and is fixed and changes, and there is a value between
and
such that
By combining these two results, you can write
If you define and as and
it follows that
By the continuity of and and the fact that and
it follows that and as and Therefore, by definition, is
differentiable.
Because and are differentiable functions of you know that both
and
approach zero as approaches zero. Moreover, because is a differentiable
function of and
you know that
where both and as So, for
from which it follows that
w
x
dx
dt
w
y
dy
dt .
dw
dt
lím
t→0
w
t
w
x
dx
dt
w
y
dy
dt
0 dx
dt
0 dy
dt
w
t
w
x
x
t
w
y
y
t
1
x
t
2
y
t
t 0
x, y → 0, 0 .
2 → 0
1
w w x x w y y 1 x 2 y,
y,
x
f
t
y
x
t,
h
g
PROOF
f
y → 0.
x → 0
2 → 0
1→ 0
y y,
y y 1
x x 1 x x
f y
f x f x x, y x f y x, y y 1 x 2 y.
z z 1 z 2 1 f x x, y x 2 f y x, y y
2 f y x x, y 1 f y x, y ,
1 f x x 1 , y f x x, y
2
1
z z 1 z 2 f x x 1 , y x f y x x, y 1 y.
z 2 f x x, y y f x x, y f y x x, y 1 y.
y
y
y
y 1
y
x
C,
B
z 1 f x x, y f x, y f x x 1 , y x.
x
x
x
x 1
x
y
B,
A
z 1 z 2 .
THEOREM 13.6 CHAIN RULE: ONE INDEPENDENT VARIABLE (PAGE 925)
Let where is a differentiable function of and If and
where and are differentiable functions of then is a differentiable
function of
and
dw
dt
w
x
dx
dt
w
y
dy
dt .
t,
w
t,
h
g
y h t ,
x
g t
y.
x
f
w f x, y ,
donde e 1 y e 2
Between and is fixed and changes. So
a value between and such that
Similarly, between and is fixed and ch
and
such that
By combining these two results, you can write
If you define and as
it follows that
By the continuity of and and the fact that
it follows that and as an
differentiable.
Because and are differentiable fun
approach zero as
approaches zero. M
function of and
you know that
where both and as
from which it follows that
dw
dt
lím
t→0
w
t
w
x
dx
dt
w
y
dy
dt
0 dx
d
w
t
w
x
x
t
w
y
y
t
1
x
t
2
y
t
x, y → 0, 0 .
2 → 0
1
w
w
y,
x
t
y
h
g
PROOF
x → 0
2 → 0
1→ 0
x
f y
f x f x x, y x f y x, y
z z 1 z 2 1 f x x, y x
1 f x x 1 , y f x x
2
1
z z 1 z 2 f x x 1 , y x f y x
z 2 f x x, y y f x x, y
y
y
y
y
x
C,
B
z 1 f x x, y f x, y f x x 1 , y
x
x
x
x 1
x
y
B,
A
THEOREM 13.6 CHAIN RULE: ONE INDEP
Let
where is a differentiable f
where and are differentiable f
tiable function of
and
dw
dt
w
x
dx
dt
w
y
dy
dt .
t,
h
g
y h t ,
f
w f x, y ,
a medida
que
Between and is fixed and changes. So, by the Mean Value Theorem, there is
a value between and such that
Similarly, between and is fixed and changes, and there is a value between
and
such that
By combining these two results, you can write
If you define and as and
it follows that
By the continuity of and and the fact that and
it follows that and as and Therefore, by definition, is
differentiable.
Because and are differentiable functions of you know that both
and
approach zero as approaches zero. Moreover, because is a differentiable
function of and
you know that
where both and as So, for
from which it follows that
w
x
dx
dt
w
y
dy
dt .
dw
dt
lím
t→0
w
t
w
x
dx
dt
w
y
dy
dt
0 dx
dt
0 dy
dt
w
t
w
x
x
t
w
y
y
t
1
x
t
2
y
t
t 0
x, y → 0, 0 .
2 → 0
1
w w x x w y y 1 x 2 y,
y,
x
f
t
y
x
t,
h
g
PROOF
f
y → 0.
x → 0
2 → 0
1→ 0
y y,
y y 1
x x 1 x x
f y
f x f x x, y x f y x, y y 1 x 2 y.
z z 1 z 2 1 f x x, y x 2 f y x, y y
2 f y x x, y 1 f y x, y ,
1 f x x 1 , y f x x, y
2
1
z z 1 z 2 f x x 1 , y x f y x x, y 1 y.
z 2 f x x, y y f x x, y f y x x, y 1 y.
y
y
y
y 1
y
x
C,
B
z 1 f x x, y f x, y f x x 1 , y x.
x
x
x
x 1
x
y
B,
A
z 1 z 2 .
THEOREM 13.6 CHAIN RULE: ONE INDEPENDENT VARIABLE (PAGE 925)
Let where is a differentiable function of and If and
where and are differentiable functions of then is a differentiable
function of
and
dw
dt
w
x
dx
dt
w
y
dy
dt .
t,
w
t,
h
g
y h t ,
x
g t
y.
x
f
w f x, y ,
por tanto, para
Between and is fixed and changes. So, by the Mean Value Theorem, there is
a value between and such that
Similarly, between and is fixed and changes, and there is a value between
and
such that
By combining these two results, you can write
If you define and as and
it follows that
By the continuity of and and the fact that and
it follows that and as and Therefore, by definition, is
differentiable.
Because and are differentiable functions of you know that both
and
approach zero as approaches zero. Moreover, because is a differentiable
function of and
you know that
where both and as So, for
from which it follows that
w
x
dx
dt
w
y
dy
dt .
dw
dt
lím
t→0
w
t
w
x
dx
dt
w
y
dy
dt
0 dx
dt
0 dy
dt
w
t
w
x
x
t
w
y
y
t
1
x
t
2
y
t
t 0
x, y → 0, 0 .
2 → 0
1
w w x x w y y 1 x 2 y,
y,
x
f
t
y
x
t,
h
g
PROOF
f
y → 0.
x → 0
2 → 0
1→ 0
y y,
y y 1
x x 1 x x
f y
f x f x x, y x f y x, y y 1 x 2 y.
z z 1 z 2 1 f x x, y x 2 f y x, y y
2 f y x x, y 1 f y x, y ,
1 f x x 1 , y f x x, y
2
1
z z 1 z 2 f x x 1 , y x f y x x, y 1 y.
z 2 f x x, y y f x x, y f y x x, y 1 y.
y
y
y
y 1
y
x
C,
B
z 1 f x x, y f x, y f x x 1 , y x.
x
x
x
x 1
x
y
B,
A
z 1 z 2 .
THEOREM 13.6 CHAIN RULE: ONE INDEPENDENT VARIABLE (PAGE 925)
Let where is a differentiable function of and If and
where and are differentiable functions of then is a differentiable
function of
and
dw
dt
w
x
dx
dt
w
y
dy
dt .
t,
w
t,
h
g
y h t ,
x
g t
y.
x
f
w f x, y ,
de lo que se deduce que
Between and is fixed and changes. So, by the Mean Value Theorem, there is
a value between and such that
Similarly, between and is fixed and changes, and there is a value between
and
such that
By combining these two results, you can write
If you define and as and
it follows that
By the continuity of and and the fact that and
it follows that and as and Therefore, by definition, is
differentiable.
Because and are differentiable functions of you know that both
and
approach zero as approaches zero. Moreover, because is a differentiable
function of and
you know that
where both and as So, for
from which it follows that
w
x
dx
dt
w
y
dy
dt .
dw
dt
lím
t→0
w
t
w
x
dx
dt
w
y
dy
dt
0 dx
dt
0 dy
dt
w
t
w
x
x
t
w
y
y
t
1
x
t
2
y
t
t 0
x, y → 0, 0 .
2 → 0
1
w w x x w y y 1 x 2 y,
y,
x
f
t
y
x
t,
h
g
PROOF
f
y → 0.
x → 0
2 → 0
1→ 0
y y,
y y 1
x x 1 x x
f y
f x f x x, y x f y x, y y 1 x 2 y.
z z 1 z 2 1 f x x, y x 2 f y x, y y
2 f y x x, y 1 f y x, y ,
1 f x x 1 , y f x x, y
2
1
z z 1 z 2 f x x 1 , y x f y x x, y 1 y.
z 2 f x x, y y f x x, y f y x x, y 1 y.
y
y
y
y 1
y
x
C,
B
z 1 f x x, y f x, y f x x 1 , y x.
x
x
x
x 1
x
y
B,
A
z 1 z 2 .
THEOREM 13.6 CHAIN RULE: ONE INDEPENDENT VARIABLE (PAGE 925)
Let where is a differentiable function of and If and
where and are differentiable functions of then is a differentiable
function of
and
dw
dt
w
x
dx
dt
w
y
dy
dt .
t,
w
t,
h
g
y h t ,
x
g t
y.
x
f
w f x, y ,
Between and is fixed and changes. So, by the Mean Value Theorem, there is
a value between and such that
Similarly, between and is fixed and changes, and there is a value between
and
such that
By combining these two results, you can write
If you define and as and
it follows that
By the continuity of and and the fact that and
it follows that and as and Therefore, by definition, is
differentiable.
Because and are differentiable functions of you know that both
and
approach zero as approaches zero. Moreover, because is a differentiable
function of and
you know that
where both and as So, for
from which it follows that
w
x
dx
dt
w
y
dy
dt .
dw
dt
lím
t→0
w
t
w
x
dx
dt
w
y
dy
dt
0 dx
dt
0 dy
dt
w
t
w
x
x
t
w
y
y
t
1
x
t
2
y
t
t 0
x, y → 0, 0 .
2 → 0
1
w w x x w y y 1 x 2 y,
y,
x
f
t
y
x
t,
h
g
PROOF
f
y → 0.
x → 0
2 → 0
1→ 0
y y,
y y 1
x x 1 x x
f y
f x f x x, y x f y x, y y 1 x 2 y.
z z 1 z 2 1 f x x, y x 2 f y x, y y
2 f y x x, y 1 f y x, y ,
1 f x x 1 , y f x x, y
2
1
z z 1 z 2 f x x 1 , y x f y x x, y 1 y.
z 2 f x x, y y f x x, y f y x x, y 1 y.
y
y
y
y 1
y
x
C,
B
z 1 f x x, y f x, y f x x 1 , y x.
x
x
x
x 1
x
y
B,
A
1 2
THEOREM 13.6 CHAIN RULE: ONE INDEPENDENT VARIABLE (PAGE 925)
Let where is a differentiable function of and If and
where and are differentiable functions of then is a differentiable
function of
and
dw
dt
w
x
dx
dt
w
y
dy
dt .
t,
w
t,
h
g
y h t ,
x
g t
y.
x
f
w f x, y ,
Between and is fixed and changes. So, by the Mean Value Theorem, there is
a value between and such that
Similarly, between and is fixed and changes, and there is a value between
and
such that
By combining these two results, you can write
If you define and as and
it follows that
By the continuity of and and the fact that and
it follows that and as and Therefore, by definition, is
differentiable.
Because and are differentiable functions of you know that both
and
approach zero as approaches zero. Moreover, because is a differentiable
function of and
you know that
where both and as So, for
from which it follows that
w
x
dx
dt
w
y
dy
dt .
dw
dt
lím
t→0
w
t
w
x
dx
dt
w
y
dy
dt
0 dx
dt
0 dy
dt
w
t
w
x
x
t
w
y
y
t
1
x
t
2
y
t
t 0
x, y → 0, 0 .
2 → 0
1
w w x x w y y 1 x 2 y,
y,
x
f
t
y
x
t,
h
g
PROOF
f
y → 0.
x → 0
2 → 0
1→ 0
y y,
y y 1
x x 1 x x
f y
f x f x x, y x f y x, y y 1 x 2 y.
z z 1 z 2 1 f x x, y x 2 f y x, y y
2 f y x x, y 1 f y x, y ,
1 f x x 1 , y f x x, y
2
1
z z 1 z 2 f x x 1 , y x f y x x, y 1 y.
z 2 f x x, y y f x x, y f y x x, y 1 y.
y
y
y
y 1
y
x
C,
B
z 1 f x x, y f x, y f x x 1 , y x.
x
x
x
x 1
x
y
B,
A
z 1 z 2 .
THEOREM 13.6 CHAIN RULE: ONE INDEPENDENT VARIABLE (PAGE 925)
Let where is a differentiable function of and If and
where and are differentiable functions of then is a differentiable
function of
and
dw
dt
w
x
dx
dt
w
y
dy
dt .
t,
w
t,
h
g
y h t ,
x
g t
y.
x
f
w f x, y ,
Between and is fixed and changes. So, by the Mean Value Theorem, there is
a value between and such that
Similarly, between and is fixed and changes, and there is a value between
and
such that
By combining these two results, you can write
If you define and as and
it follows that
By the continuity of and and the fact that and
it follows that and as and Therefore, by definition, is
differentiable.
Because and are differentiable functions of you know that both
and
approach zero as approaches zero. Moreover, because is a differentiable
function of and
you know that
where both and as So, for
from which it follows that
w
x
dx
dt
w
y
dy
dt .
dw
dt
lím
t→0
w
t
w
x
dx
dt
w
y
dy
dt
0 dx
dt
0 dy
dt
w
t
w
x
x
t
w
y
y
t
1
x
t
2
y
t
t 0
x, y → 0, 0 .
2 → 0
1
w w x x w y y 1 x 2 y,
y,
x
f
t
y
x
t,
h
g
PROOF
f
y → 0.
x → 0
2 → 0
1→ 0
y y,
y y 1
x x 1 x x
f y
f x f x x, y x f y x, y y 1 x 2 y.
z z 1 z 2 1 f x x, y x 2 f y x, y y
2 f y x x, y 1 f y x, y ,
1 f x x 1 , y f x x, y
2
1
z z 1 z 2 f x x 1 , y x f y x x, y 1 y.
z 2 f x x, y y f x x, y f y x x, y 1 y.
y
y
y
y 1
y
x
C,
B
z 1 f x x, y f x, y f x x 1 , y x.
x
x
x
x 1
x
y
B,
A
z 1 z 2 .
THEOREM 13.6 CHAIN RULE: ONE INDEPENDENT VARIABLE (PAGE 925)
Let where is a differentiable function of and If and
where and are differentiable functions of then is a differentiable
function of
and
dw
dt
w
x
dx
dt
w
y
dy
dt .
t,
w
t,
h
g
y h t ,
x
g t
y.
x
f
w f x, y ,
Between and is fixed and changes. So, by the Mean Value Theorem, there is
a value between and such that
Similarly, between and is fixed and changes, and there is a value between
and
such that
By combining these two results, you can write
If you define and as and
it follows that
By the continuity of and and the fact that and
it follows that and as and Therefore, by definition, is
differentiable.
Because and are differentiable functions of you know that both
and
approach zero as approaches zero. Moreover, because is a differentiable
function of and
you know that
where both and as So, for
from which it follows that
w
x
dx
dt
w
y
dy
dt .
dw
dt
lím
t→0
w
t
w
x
dx
dt
w
y
dy
dt
0 dx
dt
0 dy
dt
w
t
w
x
x
t
w
y
y
t
1
x
t
2
y
t
t 0
x, y → 0, 0 .
2 → 0
1
w w x x w y y 1 x 2 y,
y,
x
f
t
y
x
t,
h
g
PROOF
f
y → 0.
x → 0
2 → 0
1→ 0
y y,
y y 1
x x 1 x x
f y
f x f x x, y x f y x, y y 1 x 2 y.
z z 1 z 2 1 f x x, y x 2 f y x, y y
2 f y x x, y 1 f y x, y ,
1 f x x 1 , y f x x, y
2
1
z z 1 z 2 f x x 1 , y x f y x x, y 1 y.
z 2 f x x, y y f x x, y f y x x, y 1 y.
y
y
y
y 1
y
x
C,
B
z 1 f x x, y f x, y f x x 1 , y x.
x
x
x
x 1
x
y
B,
A
z 1 z 2 .
THEOREM 13.6 CHAIN RULE: ONE INDEPENDENT VARIABLE (PAGE 925)
Let where is a differentiable function of and If and
where and are differentiable functions of then is a differentiable
function of
and
dw
dt
w
x
dx
dt
w
y
dy
dt .
t,
w
t,
h
g
y h t ,
x
g t
y.
x
f
w f x, y ,
Between and is fixed and changes. So, by the Mean Value Theorem, there is
a value between and such that
Similarly, between and is fixed and changes, and there is a value between
and
such that
By combining these two results, you can write
If you define and as and
it follows that
By the continuity of and and the fact that and
it follows that and as and Therefore, by definition, is
differentiable.
Because and are differentiable functions of you know that both
and
approach zero as approaches zero. Moreover, because is a differentiable
function of and
you know that
where both and as So, for
from which it follows that
w
x
dx
dt
w
y
dy
dt .
dw
dt
lím
t→0
w
t
w
x
dx
dt
w
y
dy
dt
0 dx
dt
0 dy
dt
w
t
w
x
x
t
w
y
y
t
1
x
t
2
y
t
t 0
x, y → 0, 0 .
2 → 0
1
w w x x w y y 1 x 2 y,
y,
x
f
t
y
x
t,
h
g
PROOF
f
y → 0.
x → 0
2 → 0
1→ 0
y y,
y y 1
x x 1 x x
f y
f x f x x, y x f y x, y y 1 x 2 y.
z z 1 z 2 1 f x x, y x 2 f y x, y y
2 f y x x, y 1 f y x, y ,
1 f x x 1 , y f x x, y
2
1
z z 1 z 2 f x x 1 , y x f y x x, y 1 y.
z 2 f x x, y y f x x, y f y x x, y 1 y.
y
y
y
y 1
y
x
C,
B
z 1 f x x, y f x, y f x x 1 , y x.
x
x
x
x 1
x
y
B,
A
z 1 z 2 .
THEOREM 13.6 CHAIN RULE: ONE INDEPENDENT VARIABLE (PAGE 925)
Let where is a differentiable function of and If and
where and are differentiable functions of then is a differentiable
function of
and
dw
dt
w
x
dx
dt
w
y
dy
dt .
t,
w
t,
h
g
y h t ,
x
g t
y.
x
f
w f x, y ,