<s>
In	O
mathematics	O
,	O
the	O
derivative	B-Algorithm
of	I-Algorithm
a	I-Algorithm
function	I-Algorithm
of	O
a	O
real	O
variable	O
measures	O
the	O
sensitivity	O
to	O
change	O
of	O
the	O
function	O
value	O
(	O
output	O
value	O
)	O
with	O
respect	O
to	O
a	O
change	O
in	O
its	O
argument	O
(	O
input	O
value	O
)	O
.	O
</s>
<s>
Derivatives	B-Algorithm
are	O
a	O
fundamental	O
tool	O
of	O
calculus	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
derivative	B-Algorithm
of	O
the	O
position	O
of	O
a	O
moving	O
object	O
with	O
respect	O
to	O
time	O
is	O
the	O
object	O
's	O
velocity	O
:	O
this	O
measures	O
how	O
quickly	O
the	O
position	O
of	O
the	O
object	O
changes	O
when	O
time	O
advances	O
.	O
</s>
<s>
The	O
derivative	B-Algorithm
of	I-Algorithm
a	I-Algorithm
function	I-Algorithm
of	O
a	O
single	O
variable	O
at	O
a	O
chosen	O
input	O
value	O
,	O
when	O
it	O
exists	O
,	O
is	O
the	O
slope	O
of	O
the	O
tangent	O
line	O
to	O
the	O
graph	B-Application
of	I-Application
the	I-Application
function	I-Application
at	O
that	O
point	O
.	O
</s>
<s>
The	O
tangent	O
line	O
is	O
the	O
best	O
linear	B-Algorithm
approximation	I-Algorithm
of	O
the	O
function	O
near	O
that	O
input	O
value	O
.	O
</s>
<s>
For	O
this	O
reason	O
,	O
the	O
derivative	B-Algorithm
is	O
often	O
described	O
as	O
the	O
"	O
instantaneous	B-Algorithm
rate	I-Algorithm
of	I-Algorithm
change	I-Algorithm
"	O
,	O
the	O
ratio	O
of	O
the	O
instantaneous	O
change	O
in	O
the	O
dependent	O
variable	O
to	O
that	O
of	O
the	O
independent	O
variable	O
.	O
</s>
<s>
Derivatives	B-Algorithm
can	O
be	O
generalized	O
to	O
functions	O
of	O
several	O
real	O
variables	O
.	O
</s>
<s>
In	O
this	O
generalization	O
,	O
the	O
derivative	B-Algorithm
is	O
reinterpreted	O
as	O
a	O
linear	B-Architecture
transformation	I-Architecture
whose	O
graph	O
is	O
(	O
after	O
an	O
appropriate	O
translation	O
)	O
the	O
best	O
linear	B-Algorithm
approximation	I-Algorithm
to	O
the	O
graph	O
of	O
the	O
original	O
function	O
.	O
</s>
<s>
The	O
Jacobian	O
matrix	B-Architecture
is	O
the	O
matrix	B-Architecture
that	O
represents	O
this	O
linear	B-Architecture
transformation	I-Architecture
with	O
respect	O
to	O
the	O
basis	O
given	O
by	O
the	O
choice	O
of	O
independent	O
and	O
dependent	O
variables	O
.	O
</s>
<s>
It	O
can	O
be	O
calculated	O
in	O
terms	O
of	O
the	O
partial	O
derivatives	B-Algorithm
with	O
respect	O
to	O
the	O
independent	O
variables	O
.	O
</s>
<s>
For	O
a	O
real-valued	O
function	O
of	O
several	O
variables	O
,	O
the	O
Jacobian	O
matrix	B-Architecture
reduces	O
to	O
the	O
gradient	O
vector	O
.	O
</s>
<s>
The	O
process	O
of	O
finding	O
a	O
derivative	B-Algorithm
is	O
called	O
differentiation	O
.	O
</s>
<s>
The	O
reverse	O
process	O
is	O
called	O
antidifferentiation	B-Algorithm
.	O
</s>
<s>
The	O
fundamental	O
theorem	O
of	O
calculus	O
relates	O
antidifferentiation	B-Algorithm
with	O
integration	O
.	O
</s>
<s>
where	O
the	O
vertical	O
bars	O
denote	O
the	O
absolute	O
value	O
(	O
see	O
( ε	O
,	O
δ	O
)	O
-definition	O
of	O
limit	B-Algorithm
)	O
.	O
</s>
<s>
If	O
the	O
function	O
is	O
differentiable	O
at	O
,	O
that	O
is	O
if	O
the	O
limit	B-Algorithm
exists	O
,	O
then	O
this	O
limit	B-Algorithm
is	O
called	O
the	O
derivative	B-Algorithm
of	O
at	O
,	O
and	O
denoted	O
(	O
read	O
as	O
"	O
prime	O
of	O
"	O
)	O
or	O
(	O
read	O
as	O
"	O
the	O
derivative	B-Algorithm
of	O
with	O
respect	O
to	O
at	O
"	O
or	O
"	O
by	O
(	O
or	O
over	O
)	O
at	O
"	O
)	O
;	O
see	O
,	O
below	O
.	O
</s>
<s>
cannot	O
have	O
a	O
derivative	B-Algorithm
at	O
.	O
</s>
<s>
Consequently	O
,	O
the	O
secant	O
lines	O
do	O
not	O
approach	O
any	O
single	O
slope	O
,	O
so	O
the	O
limit	B-Algorithm
of	O
the	O
difference	O
quotient	O
does	O
not	O
exist	O
.	O
</s>
<s>
In	O
summary	O
,	O
a	O
function	O
that	O
has	O
a	O
derivative	B-Algorithm
is	O
continuous	O
,	O
but	O
there	O
are	O
continuous	O
functions	O
that	O
do	O
not	O
have	O
a	O
derivative	B-Algorithm
.	O
</s>
<s>
Most	O
functions	O
that	O
occur	O
in	O
practice	O
have	O
derivatives	B-Algorithm
at	O
all	O
points	O
or	O
at	O
almost	O
every	O
point	O
.	O
</s>
<s>
In	O
1931	O
,	O
Stefan	O
Banach	O
proved	O
that	O
the	O
set	O
of	O
functions	O
that	O
have	O
a	O
derivative	B-Algorithm
at	O
some	O
point	O
is	O
a	O
meager	O
set	O
in	O
the	O
space	O
of	O
all	O
continuous	O
functions	O
.	O
</s>
<s>
Informally	O
,	O
this	O
means	O
that	O
hardly	O
any	O
random	O
continuous	O
functions	O
have	O
a	O
derivative	B-Algorithm
at	O
even	O
one	O
point	O
.	O
</s>
<s>
Let	O
be	O
a	O
function	O
that	O
has	O
a	O
derivative	B-Algorithm
at	O
every	O
point	O
in	O
its	O
domain	B-Algorithm
.	O
</s>
<s>
We	O
can	O
then	O
define	O
a	O
function	O
that	O
maps	O
every	O
point	O
to	O
the	O
value	O
of	O
the	O
derivative	B-Algorithm
of	O
at	O
.	O
</s>
<s>
This	O
function	O
is	O
written	O
and	O
is	O
called	O
the	O
derivative	B-Algorithm
function	O
or	O
the	O
derivative	B-Algorithm
of	O
.	O
</s>
<s>
Sometimes	O
has	O
a	O
derivative	B-Algorithm
at	O
most	O
,	O
but	O
not	O
all	O
,	O
points	O
of	O
its	O
domain	B-Algorithm
.	O
</s>
<s>
The	O
function	O
whose	O
value	O
at	O
equals	O
whenever	O
is	O
defined	O
and	O
elsewhere	O
is	O
undefined	O
is	O
also	O
called	O
the	O
derivative	B-Algorithm
of	O
.	O
</s>
<s>
It	O
is	O
still	O
a	O
function	O
,	O
but	O
its	O
domain	B-Algorithm
may	O
be	O
smaller	O
than	O
the	O
domain	B-Algorithm
of	O
.	O
</s>
<s>
Using	O
this	O
idea	O
,	O
differentiation	O
becomes	O
a	O
function	O
of	O
functions	O
:	O
The	O
derivative	B-Algorithm
is	O
an	O
operator	O
whose	O
domain	B-Algorithm
is	O
the	O
set	O
of	O
all	O
functions	O
that	O
have	O
derivatives	B-Algorithm
at	O
every	O
point	O
of	O
their	O
domain	B-Algorithm
and	O
whose	O
range	O
is	O
a	O
set	O
of	O
functions	O
.	O
</s>
<s>
By	O
the	O
definition	O
of	O
the	O
derivative	B-Algorithm
function	O
,	O
.	O
</s>
<s>
Let	O
be	O
a	O
differentiable	O
function	O
,	O
and	O
let	O
be	O
its	O
derivative	B-Algorithm
.	O
</s>
<s>
The	O
derivative	B-Algorithm
of	O
(	O
if	O
it	O
has	O
one	O
)	O
is	O
written	O
and	O
is	O
called	O
the	O
second	B-Algorithm
derivative	I-Algorithm
of	O
.	O
</s>
<s>
Similarly	O
,	O
the	O
derivative	B-Algorithm
of	O
the	O
second	B-Algorithm
derivative	I-Algorithm
,	O
if	O
it	O
exists	O
,	O
is	O
written	O
and	O
is	O
called	O
the	O
third	O
derivative	B-Algorithm
of	O
.	O
</s>
<s>
Continuing	O
this	O
process	O
,	O
one	O
can	O
define	O
,	O
if	O
it	O
exists	O
,	O
the	O
th	O
derivative	B-Algorithm
as	O
the	O
derivative	B-Algorithm
of	O
the	O
th	O
derivative	B-Algorithm
.	O
</s>
<s>
These	O
repeated	O
derivatives	B-Algorithm
are	O
called	O
higher-order	O
derivatives	B-Algorithm
.	O
</s>
<s>
The	O
th	O
derivative	B-Algorithm
is	O
also	O
called	O
the	O
derivative	B-Algorithm
of	O
order	O
(	O
or	O
th-order	O
derivative	B-Algorithm
:	O
first-	O
,	O
second-	O
,	O
third-order	O
derivative	B-Algorithm
,	O
etc	O
.	O
)	O
</s>
<s>
If	O
represents	O
the	O
position	O
of	O
an	O
object	O
at	O
time	O
,	O
then	O
the	O
higher-order	O
derivatives	B-Algorithm
of	O
have	O
specific	O
interpretations	O
in	O
physics	O
.	O
</s>
<s>
The	O
first	B-Algorithm
derivative	I-Algorithm
of	O
is	O
the	O
object	O
's	O
velocity	O
.	O
</s>
<s>
The	O
second	B-Algorithm
derivative	I-Algorithm
of	O
is	O
the	O
acceleration	O
.	O
</s>
<s>
The	O
third	O
derivative	B-Algorithm
of	O
is	O
the	O
jerk	O
.	O
</s>
<s>
And	O
finally	O
,	O
the	O
fourth	O
through	O
sixth	O
derivatives	B-Algorithm
of	O
are	O
snap	O
,	O
crackle	O
,	O
and	O
pop	O
;	O
most	O
applicable	O
to	O
astrophysics	O
.	O
</s>
<s>
A	O
function	O
need	O
not	O
have	O
a	O
derivative	B-Algorithm
(	O
for	O
example	O
,	O
if	O
it	O
is	O
not	O
continuous	O
)	O
.	O
</s>
<s>
Similarly	O
,	O
even	O
if	O
does	O
have	O
a	O
derivative	B-Algorithm
,	O
it	O
may	O
not	O
have	O
a	O
second	B-Algorithm
derivative	I-Algorithm
.	O
</s>
<s>
{{	O
math|	O
f(x )	O
}}	O
is	O
twice	O
the	O
absolute	O
value	O
function	O
at	O
,	O
and	O
it	O
does	O
not	O
have	O
a	O
derivative	B-Algorithm
at	O
zero	O
.	O
</s>
<s>
Similar	O
examples	O
show	O
that	O
a	O
function	O
can	O
have	O
a	O
th	O
derivative	B-Algorithm
for	O
each	O
non-negative	O
integer	O
but	O
not	O
a	O
th	O
derivative	B-Algorithm
.	O
</s>
<s>
A	O
function	O
that	O
has	O
successive	O
derivatives	B-Algorithm
is	O
called	O
times	O
differentiable	O
.	O
</s>
<s>
If	O
in	O
addition	O
the	O
th	O
derivative	B-Algorithm
is	O
continuous	O
,	O
then	O
the	O
function	O
is	O
said	O
to	O
be	O
of	O
differentiability	O
class	O
.	O
</s>
<s>
(	O
This	O
is	O
a	O
stronger	O
condition	O
than	O
having	O
derivatives	B-Algorithm
,	O
as	O
shown	O
by	O
the	O
second	O
example	O
of	O
.	O
)	O
</s>
<s>
A	O
function	O
that	O
has	O
infinitely	O
many	O
derivatives	B-Algorithm
is	O
called	O
infinitely	O
differentiable	O
or	O
smooth	O
.	O
</s>
<s>
All	O
of	O
its	O
subsequent	O
derivatives	B-Algorithm
are	O
identically	O
zero	O
.	O
</s>
<s>
The	O
derivatives	B-Algorithm
of	O
a	O
function	O
at	O
a	O
point	O
provide	O
polynomial	O
approximations	O
to	O
that	O
function	O
near	O
.	O
</s>
<s>
A	O
point	O
where	O
the	O
second	B-Algorithm
derivative	I-Algorithm
of	O
a	O
function	O
changes	O
sign	O
is	O
called	O
an	O
inflection	O
point	O
.	O
</s>
<s>
At	O
an	O
inflection	O
point	O
,	O
the	O
second	B-Algorithm
derivative	I-Algorithm
may	O
be	O
zero	O
,	O
as	O
in	O
the	O
case	O
of	O
the	O
inflection	O
point	O
of	O
the	O
function	O
given	O
by	O
,	O
or	O
it	O
may	O
fail	O
to	O
exist	O
,	O
as	O
in	O
the	O
case	O
of	O
the	O
inflection	O
point	O
of	O
the	O
function	O
given	O
by	O
.	O
</s>
<s>
for	O
the	O
-th	O
derivative	B-Algorithm
of	O
.	O
</s>
<s>
These	O
are	O
abbreviations	O
for	O
multiple	O
applications	O
of	O
the	O
derivative	B-Algorithm
operator	O
.	O
</s>
<s>
With	O
Leibniz	O
's	O
notation	O
,	O
we	O
can	O
write	O
the	O
derivative	B-Algorithm
of	O
at	O
the	O
point	O
in	O
two	O
different	O
ways	O
:	O
</s>
<s>
Sometimes	O
referred	O
to	O
as	O
prime	O
notation	O
,	O
one	O
of	O
the	O
most	O
common	O
modern	O
notations	O
for	O
differentiation	O
is	O
due	O
to	O
Joseph-Louis	O
Lagrange	O
and	O
uses	O
the	O
prime	O
mark	O
,	O
so	O
that	O
the	O
derivative	B-Algorithm
of	I-Algorithm
a	I-Algorithm
function	I-Algorithm
is	O
denoted	O
.	O
</s>
<s>
To	O
denote	O
the	O
number	O
of	O
derivatives	B-Algorithm
beyond	O
this	O
point	O
,	O
some	O
authors	O
use	O
Roman	O
numerals	O
in	O
superscript	O
,	O
whereas	O
others	O
place	O
the	O
number	O
in	O
parentheses	O
:	O
</s>
<s>
The	O
latter	O
notation	O
generalizes	O
to	O
yield	O
the	O
notation	O
for	O
the	O
nth	O
derivative	B-Algorithm
of	O
–	O
this	O
notation	O
is	O
most	O
useful	O
when	O
we	O
wish	O
to	O
talk	O
about	O
the	O
derivative	B-Algorithm
as	O
being	O
a	O
function	O
itself	O
,	O
as	O
in	O
this	O
case	O
the	O
Leibniz	O
notation	O
can	O
become	O
cumbersome	O
.	O
</s>
<s>
Newton	O
's	O
notation	O
for	O
differentiation	O
,	O
also	O
called	O
the	O
dot	O
notation	O
,	O
places	O
a	O
dot	O
over	O
the	O
function	O
name	O
to	O
represent	O
a	O
time	O
derivative	B-Algorithm
.	O
</s>
<s>
denote	O
,	O
respectively	O
,	O
the	O
first	O
and	O
second	B-Algorithm
derivatives	I-Algorithm
of	O
.	O
</s>
<s>
This	O
notation	O
is	O
used	O
exclusively	O
for	O
derivatives	B-Algorithm
with	O
respect	O
to	O
time	O
or	O
arc	O
length	O
.	O
</s>
<s>
It	O
is	O
typically	O
used	O
in	O
differential	O
equations	O
in	O
physics	O
and	O
differential	B-Language
geometry	I-Language
.	O
</s>
<s>
The	O
dot	O
notation	O
,	O
however	O
,	O
becomes	O
unmanageable	O
for	O
high-order	O
derivatives	B-Algorithm
(	O
order	O
4	O
or	O
more	O
)	O
and	O
cannot	O
deal	O
with	O
multiple	O
independent	O
variables	O
.	O
</s>
<s>
Euler	O
's	O
notation	O
uses	O
a	O
differential	O
operator	O
,	O
which	O
is	O
applied	O
to	O
a	O
function	O
to	O
give	O
the	O
first	B-Algorithm
derivative	I-Algorithm
.	O
</s>
<s>
The	O
th	O
derivative	B-Algorithm
is	O
denoted	O
.	O
</s>
<s>
Euler	O
's	O
notation	O
is	O
useful	O
for	O
stating	O
and	O
solving	O
linear	B-Architecture
differential	O
equations	O
.	O
</s>
<s>
The	O
derivative	B-Algorithm
of	I-Algorithm
a	I-Algorithm
function	I-Algorithm
can	O
,	O
in	O
principle	O
,	O
be	O
computed	O
from	O
the	O
definition	O
by	O
considering	O
the	O
difference	O
quotient	O
,	O
and	O
computing	O
its	O
limit	B-Algorithm
.	O
</s>
<s>
In	O
practice	O
,	O
once	O
the	O
derivatives	B-Algorithm
of	O
a	O
few	O
simple	O
functions	O
are	O
known	O
,	O
the	O
derivatives	B-Algorithm
of	O
other	O
functions	O
are	O
more	O
easily	O
computed	O
using	O
rules	O
for	O
obtaining	O
derivatives	B-Algorithm
of	O
more	O
complicated	O
functions	O
from	O
simpler	O
ones	O
.	O
</s>
<s>
Here	O
are	O
the	O
rules	O
for	O
the	O
derivatives	B-Algorithm
of	O
the	O
most	O
common	O
basic	O
functions	O
,	O
where	O
a	O
is	O
a	O
real	O
number	O
.	O
</s>
<s>
Derivatives	B-Algorithm
of	O
powers	O
:	O
</s>
<s>
Here	O
are	O
some	O
of	O
the	O
most	O
basic	O
rules	O
for	O
deducing	O
the	O
derivative	B-Algorithm
of	O
a	O
compound	B-Application
function	I-Application
from	O
derivatives	B-Algorithm
of	O
basic	O
functions	O
.	O
</s>
<s>
The	O
known	O
derivatives	B-Algorithm
of	O
the	O
elementary	O
functions	O
x2	O
,	O
x4	O
,	O
sin(x )	O
,	O
ln(x )	O
and	O
,	O
as	O
well	O
as	O
the	O
constant	O
7	O
,	O
were	O
also	O
used	O
.	O
</s>
<s>
Relative	O
to	O
a	O
hyperreal	O
extension	O
of	O
the	O
real	O
numbers	O
,	O
the	O
derivative	B-Algorithm
of	O
a	O
real	O
function	O
at	O
a	O
real	O
point	O
can	O
be	O
defined	O
as	O
the	O
shadow	O
of	O
the	O
quotient	O
for	O
infinitesimal	O
,	O
where	O
.	O
</s>
<s>
Here	O
the	O
derivative	B-Algorithm
is	O
said	O
to	O
exist	O
if	O
the	O
shadow	O
is	O
independent	O
of	O
the	O
infinitesimal	O
chosen	O
.	O
</s>
<s>
The	O
coordinate	O
functions	O
are	O
real	O
valued	O
functions	O
,	O
so	O
the	O
above	O
definition	O
of	O
derivative	B-Algorithm
applies	O
to	O
them	O
.	O
</s>
<s>
The	O
derivative	B-Algorithm
of	O
y(t )	O
is	O
defined	O
to	O
be	O
the	O
vector	O
,	O
called	O
the	O
tangent	O
vector	O
,	O
whose	O
coordinates	O
are	O
the	O
derivatives	B-Algorithm
of	O
the	O
coordinate	O
functions	O
.	O
</s>
<s>
if	O
the	O
limit	B-Algorithm
exists	O
.	O
</s>
<s>
If	O
the	O
derivative	B-Algorithm
of	O
y	O
exists	O
for	O
every	O
value	O
of	O
t	O
,	O
then	O
y′	O
is	O
another	O
vector-valued	O
function	O
.	O
</s>
<s>
This	O
generalization	O
is	O
useful	O
,	O
for	O
example	O
,	O
if	O
y(t )	O
is	O
the	O
position	O
vector	O
of	O
a	O
particle	O
at	O
time	O
t	O
;	O
then	O
the	O
derivative	B-Algorithm
y′( t	O
)	O
is	O
the	O
velocity	O
vector	O
of	O
the	O
particle	O
at	O
time	O
t	O
.	O
</s>
<s>
Consequently	O
,	O
the	O
definition	O
of	O
the	O
derivative	B-Algorithm
for	O
a	O
function	O
of	O
one	O
variable	O
applies	O
:	O
</s>
<s>
Assembling	O
the	O
derivatives	B-Algorithm
together	O
into	O
a	O
function	O
gives	O
a	O
function	O
that	O
describes	O
the	O
variation	O
of	O
f	O
in	O
the	O
y	O
direction	O
:	O
</s>
<s>
This	O
is	O
the	O
partial	O
derivative	B-Algorithm
of	O
f	O
with	O
respect	O
to	O
y	O
.	O
</s>
<s>
Here	O
∂	O
is	O
a	O
rounded	O
d	O
called	O
the	O
partial	O
derivative	B-Algorithm
symbol	O
.	O
</s>
<s>
In	O
general	O
,	O
the	O
partial	O
derivative	B-Algorithm
of	I-Algorithm
a	I-Algorithm
function	I-Algorithm
in	O
the	O
direction	O
xi	O
at	O
the	O
point	O
(	O
a1	O
,	O
...	O
,	O
an	O
)	O
is	O
defined	O
to	O
be	O
:	O
</s>
<s>
This	O
expression	O
also	O
shows	O
that	O
the	O
computation	O
of	O
partial	O
derivatives	B-Algorithm
reduces	O
to	O
the	O
computation	O
of	O
one-variable	O
derivatives	B-Algorithm
.	O
</s>
<s>
If	O
is	O
differentiable	O
at	O
every	O
point	O
in	O
some	O
domain	B-Algorithm
,	O
then	O
the	O
gradient	O
is	O
a	O
vector-valued	O
function	O
that	O
maps	O
the	O
point	O
to	O
the	O
vector	O
.	O
</s>
<s>
If	O
f	O
is	O
a	O
real-valued	O
function	O
on	O
Rn	O
,	O
then	O
the	O
partial	O
derivatives	B-Algorithm
of	O
f	O
measure	O
its	O
variation	O
in	O
the	O
direction	O
of	O
the	O
coordinate	O
axes	O
.	O
</s>
<s>
For	O
example	O
,	O
if	O
f	O
is	O
a	O
function	O
of	O
x	O
and	O
y	O
,	O
then	O
its	O
partial	O
derivatives	B-Algorithm
measure	O
the	O
variation	O
in	O
f	O
in	O
the	O
x	O
direction	O
and	O
the	O
y	O
direction	O
.	O
</s>
<s>
These	O
are	O
measured	O
using	O
directional	O
derivatives	B-Algorithm
.	O
</s>
<s>
In	O
some	O
cases	O
it	O
may	O
be	O
easier	O
to	O
compute	O
or	O
estimate	O
the	O
directional	O
derivative	B-Algorithm
after	O
changing	O
the	O
length	O
of	O
the	O
vector	O
.	O
</s>
<s>
Often	O
this	O
is	O
done	O
to	O
turn	O
the	O
problem	O
into	O
the	O
computation	O
of	O
a	O
directional	O
derivative	B-Algorithm
in	O
the	O
direction	O
of	O
a	O
unit	O
vector	O
.	O
</s>
<s>
This	O
is	O
λ	O
times	O
the	O
difference	O
quotient	O
for	O
the	O
directional	O
derivative	B-Algorithm
of	O
f	O
with	O
respect	O
to	O
u	O
.	O
</s>
<s>
Furthermore	O
,	O
taking	O
the	O
limit	B-Algorithm
as	O
h	O
tends	O
to	O
zero	O
is	O
the	O
same	O
as	O
taking	O
the	O
limit	B-Algorithm
as	O
k	O
tends	O
to	O
zero	O
because	O
h	O
and	O
k	O
are	O
multiples	O
of	O
each	O
other	O
.	O
</s>
<s>
Because	O
of	O
this	O
rescaling	O
property	O
,	O
directional	O
derivatives	B-Algorithm
are	O
frequently	O
considered	O
only	O
for	O
unit	O
vectors	O
.	O
</s>
<s>
If	O
all	O
the	O
partial	O
derivatives	B-Algorithm
of	O
f	O
exist	O
and	O
are	O
continuous	O
at	O
x	O
,	O
then	O
they	O
determine	O
the	O
directional	O
derivative	B-Algorithm
of	O
f	O
in	O
the	O
direction	O
v	O
by	O
the	O
formula	O
:	O
</s>
<s>
This	O
is	O
a	O
consequence	O
of	O
the	O
definition	O
of	O
the	O
total	O
derivative	B-Algorithm
.	O
</s>
<s>
It	O
follows	O
that	O
the	O
directional	O
derivative	B-Algorithm
is	O
linear	B-Architecture
in	O
v	O
,	O
meaning	O
that	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
the	O
directional	O
derivative	B-Algorithm
is	O
a	O
vector	O
in	O
Rm	O
.	O
</s>
<s>
When	O
f	O
is	O
a	O
function	O
from	O
an	O
open	O
subset	O
of	O
Rn	O
to	O
Rm	O
,	O
then	O
the	O
directional	O
derivative	B-Algorithm
of	O
f	O
in	O
a	O
chosen	O
direction	O
is	O
the	O
best	O
linear	B-Algorithm
approximation	I-Algorithm
to	O
f	O
at	O
that	O
point	O
and	O
in	O
that	O
direction	O
.	O
</s>
<s>
But	O
when	O
,	O
no	O
single	O
directional	O
derivative	B-Algorithm
can	O
give	O
a	O
complete	O
picture	O
of	O
the	O
behavior	O
of	O
f	O
.	O
The	O
total	O
derivative	B-Algorithm
gives	O
a	O
complete	O
picture	O
by	O
considering	O
all	O
directions	O
at	O
once	O
.	O
</s>
<s>
That	O
is	O
,	O
for	O
any	O
vector	O
v	O
starting	O
at	O
a	O
,	O
the	O
linear	B-Algorithm
approximation	I-Algorithm
formula	O
holds	O
:	O
</s>
<s>
Just	O
like	O
the	O
single-variable	O
derivative	B-Algorithm
,	O
is	O
chosen	O
so	O
that	O
the	O
error	O
in	O
this	O
approximation	O
is	O
as	O
small	O
as	O
possible	O
.	O
</s>
<s>
If	O
n	O
and	O
m	O
are	O
both	O
one	O
,	O
then	O
the	O
derivative	B-Algorithm
is	O
a	O
number	O
and	O
the	O
expression	O
is	O
the	O
product	O
of	O
two	O
numbers	O
.	O
</s>
<s>
For	O
the	O
linear	B-Algorithm
approximation	I-Algorithm
formula	O
to	O
make	O
sense	O
,	O
must	O
be	O
a	O
function	O
that	O
sends	O
vectors	O
in	O
Rn	O
to	O
vectors	O
in	O
Rm	O
,	O
and	O
must	O
denote	O
this	O
function	O
evaluated	O
at	O
v	O
.	O
</s>
<s>
If	O
we	O
assume	O
that	O
v	O
is	O
small	O
and	O
that	O
the	O
derivative	B-Algorithm
varies	O
continuously	O
in	O
a	O
,	O
then	O
is	O
approximately	O
equal	O
to	O
,	O
and	O
therefore	O
the	O
right-hand	O
side	O
is	O
approximately	O
zero	O
.	O
</s>
<s>
The	O
left-hand	O
side	O
can	O
be	O
rewritten	O
in	O
a	O
different	O
way	O
using	O
the	O
linear	B-Algorithm
approximation	I-Algorithm
formula	O
with	O
substituted	O
for	O
v	O
.	O
The	O
linear	B-Algorithm
approximation	I-Algorithm
formula	O
implies	O
:	O
</s>
<s>
This	O
suggests	O
that	O
is	O
a	O
linear	B-Architecture
transformation	I-Architecture
from	O
the	O
vector	O
space	O
Rn	O
to	O
the	O
vector	O
space	O
Rm	O
.	O
</s>
<s>
In	O
fact	O
,	O
it	O
is	O
possible	O
to	O
make	O
this	O
a	O
precise	O
derivation	B-Algorithm
by	O
measuring	O
the	O
error	O
in	O
the	O
approximations	O
.	O
</s>
<s>
Assume	O
that	O
the	O
error	O
in	O
these	O
linear	B-Algorithm
approximation	I-Algorithm
formula	O
is	O
bounded	O
by	O
a	O
constant	O
times	O
||v||	O
,	O
where	O
the	O
constant	O
is	O
independent	O
of	O
v	O
but	O
depends	O
continuously	O
on	O
a	O
.	O
</s>
<s>
In	O
particular	O
,	O
is	O
a	O
linear	B-Architecture
transformation	I-Architecture
up	O
to	O
a	O
small	O
error	O
term	O
.	O
</s>
<s>
In	O
the	O
limit	B-Algorithm
as	O
v	O
and	O
w	O
tend	O
to	O
zero	O
,	O
it	O
must	O
therefore	O
be	O
a	O
linear	B-Architecture
transformation	I-Architecture
.	O
</s>
<s>
Since	O
we	O
define	O
the	O
total	O
derivative	B-Algorithm
by	O
taking	O
a	O
limit	B-Algorithm
as	O
v	O
goes	O
to	O
zero	O
,	O
must	O
be	O
a	O
linear	B-Architecture
transformation	I-Architecture
.	O
</s>
<s>
In	O
one	O
variable	O
,	O
the	O
fact	O
that	O
the	O
derivative	B-Algorithm
is	O
the	O
best	O
linear	B-Algorithm
approximation	I-Algorithm
is	O
expressed	O
by	O
the	O
fact	O
that	O
it	O
is	O
the	O
limit	B-Algorithm
of	O
difference	O
quotients	O
.	O
</s>
<s>
In	O
particular	O
,	O
the	O
numerator	O
and	O
denominator	O
of	O
the	O
difference	O
quotient	O
are	O
not	O
even	O
in	O
the	O
same	O
vector	O
space	O
:	O
The	O
numerator	O
lies	O
in	O
the	O
codomain	O
Rm	O
while	O
the	O
denominator	O
lies	O
in	O
the	O
domain	B-Algorithm
Rn	O
.	O
</s>
<s>
Furthermore	O
,	O
the	O
derivative	B-Algorithm
is	O
a	O
linear	B-Architecture
transformation	I-Architecture
,	O
a	O
different	O
type	O
of	O
object	O
from	O
both	O
the	O
numerator	O
and	O
denominator	O
.	O
</s>
<s>
To	O
make	O
precise	O
the	O
idea	O
that	O
is	O
the	O
best	O
linear	B-Algorithm
approximation	I-Algorithm
,	O
it	O
is	O
necessary	O
to	O
adapt	O
a	O
different	O
formula	O
for	O
the	O
one-variable	O
derivative	B-Algorithm
in	O
which	O
these	O
problems	O
disappear	O
.	O
</s>
<s>
because	O
the	O
limit	B-Algorithm
of	O
a	O
function	O
tends	O
to	O
zero	O
if	O
and	O
only	O
if	O
the	O
limit	B-Algorithm
of	O
the	O
absolute	O
value	O
of	O
the	O
function	O
tends	O
to	O
zero	O
.	O
</s>
<s>
If	O
the	O
total	O
derivative	B-Algorithm
exists	O
at	O
a	O
,	O
then	O
all	O
the	O
partial	O
derivatives	B-Algorithm
and	O
directional	O
derivatives	B-Algorithm
of	O
f	O
exist	O
at	O
a	O
,	O
and	O
for	O
all	O
v	O
,	O
is	O
the	O
directional	O
derivative	B-Algorithm
of	O
f	O
in	O
the	O
direction	O
v	O
.	O
If	O
we	O
write	O
f	O
using	O
coordinate	O
functions	O
,	O
so	O
that	O
,	O
then	O
the	O
total	O
derivative	B-Algorithm
can	O
be	O
expressed	O
using	O
the	O
partial	O
derivatives	B-Algorithm
as	O
a	O
matrix	B-Architecture
.	O
</s>
<s>
This	O
matrix	B-Architecture
is	O
called	O
the	O
Jacobian	O
matrix	B-Architecture
of	O
f	O
at	O
a	O
:	O
</s>
<s>
The	O
existence	O
of	O
the	O
total	O
derivative	B-Algorithm
f′( a	O
)	O
is	O
strictly	O
stronger	O
than	O
the	O
existence	O
of	O
all	O
the	O
partial	O
derivatives	B-Algorithm
,	O
but	O
if	O
the	O
partial	O
derivatives	B-Algorithm
exist	O
and	O
are	O
continuous	O
,	O
then	O
the	O
total	O
derivative	B-Algorithm
exists	O
,	O
is	O
given	O
by	O
the	O
Jacobian	O
,	O
and	O
depends	O
continuously	O
on	O
a	O
.	O
</s>
<s>
The	O
definition	O
of	O
the	O
total	O
derivative	B-Algorithm
subsumes	O
the	O
definition	O
of	O
the	O
derivative	B-Algorithm
in	O
one	O
variable	O
.	O
</s>
<s>
That	O
is	O
,	O
if	O
f	O
is	O
a	O
real-valued	O
function	O
of	O
a	O
real	O
variable	O
,	O
then	O
the	O
total	O
derivative	B-Algorithm
exists	O
if	O
and	O
only	O
if	O
the	O
usual	O
derivative	B-Algorithm
exists	O
.	O
</s>
<s>
The	O
Jacobian	O
matrix	B-Architecture
reduces	O
to	O
a	O
1×1	O
matrix	B-Architecture
whose	O
only	O
entry	O
is	O
the	O
derivative	B-Algorithm
f′( x	O
)	O
.	O
</s>
<s>
Up	O
to	O
changing	O
variables	O
,	O
this	O
is	O
the	O
statement	O
that	O
the	O
function	O
is	O
the	O
best	O
linear	B-Algorithm
approximation	I-Algorithm
to	O
f	O
at	O
a	O
.	O
</s>
<s>
The	O
total	O
derivative	B-Algorithm
of	I-Algorithm
a	I-Algorithm
function	I-Algorithm
does	O
not	O
give	O
another	O
function	O
in	O
the	O
same	O
way	O
as	O
the	O
one-variable	O
case	O
.	O
</s>
<s>
This	O
is	O
because	O
the	O
total	O
derivative	B-Algorithm
of	O
a	O
multivariable	O
function	O
has	O
to	O
record	O
much	O
more	O
information	O
than	O
the	O
derivative	B-Algorithm
of	O
a	O
single-variable	O
function	O
.	O
</s>
<s>
Instead	O
,	O
the	O
total	O
derivative	B-Algorithm
gives	O
a	O
function	O
from	O
the	O
tangent	O
bundle	O
of	O
the	O
source	O
to	O
the	O
tangent	O
bundle	O
of	O
the	O
target	O
.	O
</s>
<s>
The	O
natural	O
analog	O
of	O
second	O
,	O
third	O
,	O
and	O
higher-order	O
total	O
derivatives	B-Algorithm
is	O
not	O
a	O
linear	B-Architecture
transformation	I-Architecture
,	O
is	O
not	O
a	O
function	O
on	O
the	O
tangent	O
bundle	O
,	O
and	O
is	O
not	O
built	O
by	O
repeatedly	O
taking	O
the	O
total	O
derivative	B-Algorithm
.	O
</s>
<s>
The	O
analog	O
of	O
a	O
higher-order	O
derivative	B-Algorithm
,	O
called	O
a	O
jet	O
,	O
cannot	O
be	O
a	O
linear	B-Architecture
transformation	I-Architecture
because	O
higher-order	O
derivatives	B-Algorithm
reflect	O
subtle	O
geometric	O
information	O
,	O
such	O
as	O
concavity	O
,	O
which	O
cannot	O
be	O
described	O
in	O
terms	O
of	O
linear	B-Architecture
data	O
such	O
as	O
vectors	O
.	O
</s>
<s>
It	O
cannot	O
be	O
a	O
function	O
on	O
the	O
tangent	O
bundle	O
because	O
the	O
tangent	O
bundle	O
only	O
has	O
room	O
for	O
the	O
base	O
space	O
and	O
the	O
directional	O
derivatives	B-Algorithm
.	O
</s>
<s>
The	O
relation	O
between	O
the	O
total	O
derivative	B-Algorithm
and	O
the	O
partial	O
derivatives	B-Algorithm
of	O
a	O
function	O
is	O
paralleled	O
in	O
the	O
relation	O
between	O
the	O
kth	O
order	O
jet	O
of	O
a	O
function	O
and	O
its	O
partial	O
derivatives	B-Algorithm
of	O
order	O
less	O
than	O
or	O
equal	O
to	O
k	O
.	O
</s>
<s>
By	O
repeatedly	O
taking	O
the	O
total	O
derivative	B-Algorithm
,	O
one	O
obtains	O
higher	O
versions	O
of	O
the	O
Fréchet	O
derivative	B-Algorithm
,	O
specialized	O
to	O
Rp	O
.	O
</s>
<s>
where	O
f(a''' )	O
is	O
identified	O
with	O
a	O
constant	O
function	O
,	O
are	O
the	O
components	O
of	O
the	O
vector	O
,	O
and	O
and	O
are	O
the	O
components	O
of	O
and	O
as	O
linear	B-Architecture
transformations	I-Architecture
.	O
</s>
<s>
The	O
concept	O
of	O
a	O
derivative	B-Algorithm
can	O
be	O
extended	O
to	O
many	O
other	O
settings	O
.	O
</s>
<s>
The	O
common	O
thread	O
is	O
that	O
the	O
derivative	B-Algorithm
of	I-Algorithm
a	I-Algorithm
function	I-Algorithm
at	O
a	O
point	O
serves	O
as	O
a	O
linear	B-Algorithm
approximation	I-Algorithm
of	O
the	O
function	O
at	O
that	O
point	O
.	O
</s>
<s>
An	O
important	O
generalization	O
of	O
the	O
derivative	B-Algorithm
concerns	O
complex	O
functions	O
of	O
complex	O
variables	O
,	O
such	O
as	O
functions	O
from	O
(	O
a	O
domain	B-Algorithm
in	O
)	O
the	O
complex	O
numbers	O
to	O
.	O
</s>
<s>
The	O
notion	O
of	O
the	O
derivative	B-Algorithm
of	O
such	O
a	O
function	O
is	O
obtained	O
by	O
replacing	O
real	O
variables	O
with	O
complex	O
variables	O
in	O
the	O
definition	O
.	O
</s>
<s>
If	O
is	O
identified	O
with	O
by	O
writing	O
a	O
complex	O
number	O
as	O
,	O
then	O
a	O
differentiable	O
function	O
from	O
to	O
is	O
certainly	O
differentiable	O
as	O
a	O
function	O
from	O
to	O
(	O
in	O
the	O
sense	O
that	O
its	O
partial	O
derivatives	B-Algorithm
all	O
exist	O
)	O
,	O
but	O
the	O
converse	O
is	O
not	O
true	O
in	O
general	O
:	O
the	O
complex	O
derivative	B-Algorithm
only	O
exists	O
if	O
the	O
real	O
derivative	B-Algorithm
is	O
complex	O
linear	B-Architecture
and	O
this	O
imposes	O
relations	O
between	O
the	O
partial	O
derivatives	B-Algorithm
called	O
the	O
Cauchy	O
–	O
Riemann	O
equations	O
–	O
see	O
holomorphic	O
functions	O
.	O
</s>
<s>
The	O
derivative	B-Algorithm
(	O
or	O
differential	O
)	O
of	O
a	O
(	O
differentiable	O
)	O
map	O
between	O
manifolds	O
,	O
at	O
a	O
point	O
in	O
,	O
is	O
then	O
a	O
linear	B-Architecture
map	I-Architecture
from	O
the	O
tangent	O
space	O
of	O
at	O
to	O
the	O
tangent	O
space	O
of	O
at	O
.	O
</s>
<s>
The	O
derivative	B-Algorithm
function	O
becomes	O
a	O
map	O
between	O
the	O
tangent	O
bundles	O
of	O
and	O
.	O
</s>
<s>
This	O
definition	O
is	O
fundamental	O
in	O
differential	B-Language
geometry	I-Language
and	O
has	O
many	O
uses	O
–	O
see	O
pushforward	O
(	O
differential	O
)	O
and	O
pullback	O
(	O
differential	B-Language
geometry	I-Language
)	O
.	O
</s>
<s>
Differentiation	O
can	O
also	O
be	O
defined	O
for	O
maps	O
between	O
infinite	O
dimensional	O
vector	O
spaces	O
such	O
as	O
Banach	O
spaces	O
and	O
Fréchet	B-Algorithm
spaces	I-Algorithm
.	O
</s>
<s>
There	O
is	O
a	O
generalization	O
both	O
of	O
the	O
directional	O
derivative	B-Algorithm
,	O
called	O
the	O
Gateaux	B-Algorithm
derivative	I-Algorithm
,	O
and	O
of	O
the	O
differential	O
,	O
called	O
the	O
Fréchet	O
derivative	B-Algorithm
.	O
</s>
<s>
One	O
deficiency	O
of	O
the	O
classical	O
derivative	B-Algorithm
is	O
that	O
very	O
many	O
functions	O
are	O
not	O
differentiable	O
.	O
</s>
<s>
Nevertheless	O
,	O
there	O
is	O
a	O
way	O
of	O
extending	O
the	O
notion	O
of	O
the	O
derivative	B-Algorithm
so	O
that	O
all	O
continuous	O
functions	O
and	O
many	O
other	O
functions	O
can	O
be	O
differentiated	O
using	O
a	O
concept	O
known	O
as	O
the	O
weak	O
derivative	B-Algorithm
.	O
</s>
<s>
The	O
properties	O
of	O
the	O
derivative	B-Algorithm
have	O
inspired	O
the	O
introduction	O
and	O
study	O
of	O
many	O
similar	O
objects	O
in	O
algebra	O
and	O
topology	O
—	O
see	O
,	O
for	O
example	O
,	O
differential	O
algebra	O
.	O
</s>
<s>
The	O
discrete	O
equivalent	O
of	O
differentiation	O
is	O
finite	B-Algorithm
differences	I-Algorithm
.	O
</s>
<s>
The	O
study	O
of	O
differential	O
calculus	O
is	O
unified	O
with	O
the	O
calculus	O
of	O
finite	B-Algorithm
differences	I-Algorithm
in	O
time	O
scale	O
calculus	O
.	O
</s>
<s>
Also	O
see	O
arithmetic	O
derivative	B-Algorithm
.	O
</s>
<s>
Calculus	O
,	O
known	O
in	O
its	O
early	O
history	O
as	O
infinitesimal	O
calculus''	O
,	O
is	O
a	O
mathematical	O
discipline	O
focused	O
on	O
limits	B-Algorithm
,	O
functions	O
,	O
derivatives	B-Algorithm
,	O
integrals	O
,	O
and	O
infinite	O
series	O
.	O
</s>
