För den fordonstekniska delen, se Differentialväxel , Differentialbroms , Torsendifferential
Differential är en term inom matematisk analys för en infinitesimal - oändligt liten - ändring i en funktion .
Låt
f
:
U
→
R
m
{\displaystyle f\colon U\rightarrow \mathbb {R} ^{m}}
U
{\displaystyle U}
öppen delmängd i
R
n
{\displaystyle \mathbb {R} ^{n}}
f
{\displaystyle f}
differentierbar [ 1]
a
∈
U
{\displaystyle a\in U}
L
{\displaystyle L}
lim
|
h
|
→
0
f
(
a
+
h
)
−
f
(
a
)
−
L
(
h
)
|
h
|
=
0
{\displaystyle \lim _{|h|\to 0}{\frac {f(a+h)-f(a)-L(h)}{|h|}}=0\ }
Den linjära avbildningen
L
{\displaystyle L}
differentialen till
f
{\displaystyle f}
a
{\displaystyle a}
d
f
a
{\displaystyle df_{a}}
Δ
a
f
(
h
)
=
f
(
a
+
h
)
−
f
(
a
)
{\displaystyle \Delta _{a}f(h)=f(a+h)-f(a)}
h
{\displaystyle h}
f
(
a
+
h
)
≈
f
(
a
)
+
d
f
a
(
h
)
{\displaystyle f(a+h)\approx f(a)+df_{a}(h)}
Matrisen hörande till differentialen betecknas
f
′
(
a
)
{\displaystyle f'(a)}
funktionalmatrisen eller jacobimatrisen .
I fallet
m
=
n
=
1
{\displaystyle m=n=1}
f
′
(
a
)
{\displaystyle f'(a)}
derivatan i
a
{\displaystyle a}
m
=
1
,
n
>
1
{\displaystyle m=1,n>1}
f
′
(
a
)
{\displaystyle f'(a)}
∇
f
(
a
)
{\displaystyle \nabla f(a)}
Riktningsderivatan ,
D
v
f
(
a
)
{\displaystyle D_{v}f(a)}
f
{\displaystyle f}
a
{\displaystyle a}
v
≠
0
{\displaystyle v\neq 0}
D
v
f
(
a
)
=
lim
t
→
0
f
(
a
+
t
v
)
−
f
(
a
)
t
{\displaystyle D_{v}f(a)=\lim _{t\to 0}{\frac {f(a+tv)-f(a)}{t}}}
En räkning ger,
D
v
f
(
a
)
=
lim
t
→
0
f
(
a
+
t
v
)
−
f
(
a
)
t
=
lim
t
→
0
f
(
a
+
t
v
)
−
f
(
a
)
−
d
f
a
(
t
v
)
+
d
f
a
(
t
v
)
t
=
{\displaystyle D_{v}f(a)=\lim _{t\to 0}{\frac {f(a+tv)-f(a)}{t}}=\lim _{t\to 0}{\frac {f(a+tv)-f(a)-df_{a}(tv)+df_{a}(tv)}{t}}=}
=
lim
t
→
0
f
(
a
+
t
v
)
−
f
(
a
)
−
d
f
a
(
t
v
)
t
+
d
f
a
(
v
)
=
0
+
d
f
a
(
v
)
=
d
f
a
(
v
)
{\displaystyle \lim _{t\to 0}{\frac {f(a+tv)-f(a)-df_{a}(tv)}{t}}+df_{a}(v)=0+df_{a}(v)=df_{a}(v)}
varför
D
v
f
(
a
)
=
d
f
a
(
v
)
{\displaystyle D_{v}f(a)=df_{a}(v)}
Riktningsderivatan kan sålunda uttryckas med differentialen; speciellt betyder detta att riktningsderivatan är linjär i
v
{\displaystyle v}
D
0
=
0
{\displaystyle D_{0}=0}
Betrakta fallet
m
=
n
=
1
{\displaystyle m=n=1}
x
{\displaystyle x}
R
→
R
{\displaystyle \mathbb {R} \rightarrow \mathbb {R} }
x
{\displaystyle x}
d
x
a
(
h
)
=
1
⋅
h
=
h
{\displaystyle dx_{a}(h)=1\cdot h=h}
f
:
R
→
R
{\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {R} }
d
f
a
(
h
)
=
f
′
(
a
)
h
{\displaystyle df_{a}(h)=f'(a)h}
d
f
a
(
h
)
=
f
′
(
a
)
d
x
a
(
h
)
{\displaystyle df_{a}(h)=f'(a)dx_{a}(h)}
Leibniz notation ,
f
′
(
a
)
=
d
f
/
d
x
{\displaystyle f'(a)=df/dx}
h
{\displaystyle h}
d
f
=
d
f
d
x
d
x
{\displaystyle df={\frac {df}{dx}}dx}
Analogt fås i fallet
m
=
1
,
n
>
1
{\displaystyle m=1,n>1}
d
f
=
∂
f
∂
x
1
d
x
1
+
…
+
∂
f
∂
x
n
d
x
n
{\displaystyle df={\frac {\partial f}{\partial x_{1}}}dx_{1}+\ldots +{\frac {\partial f}{\partial x_{n}}}dx_{n}}
Låt
f
:
R
→
R
{\displaystyle f\colon \mathbb {R} \rightarrow \mathbb {R} }
f
(
x
)
=
sin
(
x
)
{\displaystyle f(x)=\sin(x)}
f
{\displaystyle f}
a
=
π
{\displaystyle a=\pi }
f
′
(
π
)
=
cos
(
π
)
=
−
1
{\displaystyle f'(\pi )=\cos(\pi )=-1}
f
(
3
)
{\displaystyle f(3)}
h
=
3
−
π
≈
−
0.14
{\displaystyle h=3-\pi \approx -0.14}
a
=
π
{\displaystyle a=\pi }
f
(
3
)
=
f
(
π
+
(
3
−
π
)
)
=
f
(
a
+
h
)
≈
f
(
a
)
+
d
f
a
(
h
)
≈
f
(
π
)
+
(
−
1
)
⋅
(
−
0.14
)
=
0.14.
{\displaystyle f(3)=f(\pi +(3-\pi ))=f(a+h)\approx f(a)+df_{a}(h)\approx f(\pi )+(-1)\cdot (-0.14)=0.14.}
Anm. Med fem decimalers noggrannhet är
f
(
3
)
≈
0.14112
{\displaystyle f(3)\approx 0.14112}
^ Edwards, Jr., C.H. (1994). Advanced Calculus of Several Variables ISBN 978-0-486-68336-2
