<s>
which	O
when	O
taken	O
to	O
the	O
limit	B-Algorithm
as	O
h	O
approaches	O
0	O
gives	O
the	O
derivative	B-Algorithm
of	O
the	O
function	O
f	O
.	O
The	O
name	O
of	O
the	O
expression	O
stems	O
from	O
the	O
fact	O
that	O
it	O
is	O
the	O
quotient	O
of	O
the	O
difference	O
of	O
values	O
of	O
the	O
function	O
by	O
the	O
difference	O
of	O
the	O
corresponding	O
values	O
of	O
its	O
argument	O
(	O
the	O
latter	O
is	O
(	O
x	O
+	O
h	O
)	O
-	O
x	O
=	O
h	O
in	O
this	O
case	O
)	O
.	O
</s>
<s>
The	O
difference	B-Algorithm
quotient	I-Algorithm
is	O
a	O
measure	O
of	O
the	O
average	O
rate	O
of	O
change	O
of	O
the	O
function	O
over	O
an	O
interval	O
(	O
in	O
this	O
case	O
,	O
an	O
interval	O
of	O
length	O
h	O
)	O
.	O
</s>
<s>
The	O
limit	B-Algorithm
of	O
the	O
difference	B-Algorithm
quotient	I-Algorithm
(	O
i.e.	O
,	O
the	O
derivative	B-Algorithm
)	O
is	O
thus	O
the	O
instantaneous	B-Algorithm
rate	I-Algorithm
of	I-Algorithm
change	I-Algorithm
.	O
</s>
<s>
is	O
called	O
the	O
mean	O
(	O
or	O
average	O
)	O
value	O
of	O
the	O
derivative	B-Algorithm
of	O
f	O
over	O
the	O
interval	O
[	O
a	O
,	O
b ]	O
.	O
</s>
<s>
This	O
name	O
is	O
justified	O
by	O
the	O
mean	B-Algorithm
value	I-Algorithm
theorem	I-Algorithm
,	O
which	O
states	O
that	O
for	O
a	O
differentiable	O
function	O
f	O
,	O
its	O
derivative	B-Algorithm
f′	O
reaches	O
its	O
mean	O
value	O
at	O
some	O
point	O
in	O
the	O
interval	O
.	O
</s>
<s>
Geometrically	O
,	O
this	O
difference	B-Algorithm
quotient	I-Algorithm
measures	O
the	O
slope	O
of	O
the	O
secant	O
line	O
passing	O
through	O
the	O
points	O
with	O
coordinates	O
(	O
a	O
,	O
f(a )	O
)	O
and	O
(	O
b	O
,	O
f(b )	O
)	O
.	O
</s>
<s>
Difference	B-Algorithm
quotients	I-Algorithm
are	O
used	O
as	O
approximations	O
in	O
numerical	B-Algorithm
differentiation	I-Algorithm
,	O
but	O
they	O
have	O
also	O
been	O
subject	O
of	O
criticism	O
in	O
this	O
application	O
.	O
</s>
<s>
Difference	B-Algorithm
quotients	I-Algorithm
may	O
also	O
find	O
relevance	O
in	O
applications	O
involving	O
Time	O
discretization	O
,	O
where	O
the	O
width	O
of	O
the	O
time	O
step	O
is	O
used	O
for	O
the	O
value	O
of	O
h	O
.	O
</s>
<s>
The	O
difference	B-Algorithm
quotient	I-Algorithm
is	O
sometimes	O
also	O
called	O
the	O
Newton	B-Algorithm
quotient	I-Algorithm
(	O
after	O
Isaac	O
Newton	O
)	O
or	O
Fermat	B-Algorithm
's	I-Algorithm
difference	I-Algorithm
quotient	I-Algorithm
(	O
after	O
Pierre	O
de	O
Fermat	O
)	O
.	O
</s>
<s>
The	O
typical	O
notion	O
of	O
the	O
difference	B-Algorithm
quotient	I-Algorithm
discussed	O
above	O
is	O
a	O
particular	O
case	O
of	O
a	O
more	O
general	O
concept	O
.	O
</s>
<s>
Forward	B-Algorithm
difference	I-Algorithm
:	O
ΔF(P )	O
=	O
F( P	O
+	O
ΔP	O
)	O
−	O
F(P )	O
;	O
</s>
<s>
Central	B-Algorithm
difference	I-Algorithm
:	O
δF(P )	O
=	O
F( P	O
+	O
½ΔP	O
)	O
−	O
F( P	O
−	O
½ΔP	O
)	O
;	O
</s>
<s>
Backward	B-Algorithm
difference	I-Algorithm
:	O
∇	O
F(P )	O
=	O
F(P )	O
−	O
F( P	O
−	O
ΔP	O
)	O
.	O
</s>
<s>
If	O
|ΔP|	O
is	O
finite	O
(	O
meaning	O
measurable	O
)	O
,	O
then	O
ΔF(P )	O
is	O
known	O
as	O
a	O
finite	B-Algorithm
difference	I-Algorithm
,	O
with	O
specific	O
denotations	O
of	O
DP	O
and	O
DF(P )	O
;	O
</s>
<s>
If	O
|ΔP|	O
is	O
infinitesimal	O
(	O
an	O
infinitely	O
small	O
amount	O
—	O
—	O
usually	O
expressed	O
in	O
standard	O
analysis	O
as	O
a	O
limit	B-Algorithm
:	O
)	O
,	O
then	O
ΔF(P )	O
is	O
known	O
as	O
an	O
infinitesimal	O
difference	O
,	O
with	O
specific	O
denotations	O
of	O
dP	O
and	O
dF(P )	O
(	O
in	O
calculus	O
graphing	O
,	O
the	O
point	O
is	O
almost	O
exclusively	O
identified	O
as	O
"	O
x	O
"	O
and	O
F(x )	O
as	O
"	O
y	O
"	O
)	O
.	O
</s>
<s>
The	O
function	O
difference	O
divided	O
by	O
the	O
point	O
difference	O
is	O
known	O
as	O
"	O
difference	B-Algorithm
quotient	I-Algorithm
"	O
:	O
</s>
<s>
If	O
ΔP	O
is	O
infinitesimal	O
,	O
then	O
the	O
difference	B-Algorithm
quotient	I-Algorithm
is	O
a	O
derivative	B-Algorithm
,	O
otherwise	O
it	O
is	O
a	O
divided	B-Algorithm
difference	I-Algorithm
:	O
</s>
<s>
Regardless	O
if	O
ΔP	O
is	O
infinitesimal	O
or	O
finite	O
,	O
there	O
is	O
(	O
at	O
least	O
—	O
in	O
the	O
case	O
of	O
the	O
derivative	B-Algorithm
—	O
theoretically	O
)	O
a	O
point	O
range	O
,	O
where	O
the	O
boundaries	O
are	O
P±( 	O
0.5	O
)	O
ΔP	O
(	O
depending	O
on	O
the	O
orientation	O
—	O
ΔF(P )	O
,	O
δF(P )	O
or	O
∇	O
F(P )	O
)	O
:	O
</s>
<s>
Derivatives	B-Algorithm
can	O
be	O
regarded	O
as	O
functions	O
themselves	O
,	O
harboring	O
their	O
own	O
derivatives	B-Algorithm
.	O
</s>
<s>
Thus	O
each	O
function	O
is	O
home	O
to	O
sequential	O
degrees	O
(	O
"	O
higher	O
orders	O
"	O
)	O
of	O
derivation	B-Algorithm
,	O
or	O
differentiation	O
.	O
</s>
<s>
This	O
property	O
can	O
be	O
generalized	O
to	O
all	O
difference	B-Algorithm
quotients	I-Algorithm
.	O
</s>
<s>
The	O
difference	B-Algorithm
quotient	I-Algorithm
as	O
a	O
derivative	B-Algorithm
needs	O
no	O
explanation	O
,	O
other	O
than	O
to	O
point	O
out	O
that	O
,	O
since	O
P0	O
essentially	O
equals	O
P1	O
=	O
P2	O
=	O
...	O
=	O
Pń	O
(	O
as	O
the	O
differences	O
are	O
infinitesimal	O
)	O
,	O
the	O
Leibniz	O
notation	O
and	O
derivative	B-Algorithm
expressions	O
do	O
not	O
distinguish	O
P	O
to	O
P0	O
or	O
Pń	O
:	O
</s>
<s>
There	O
are	O
other	O
derivative	B-Algorithm
notations	O
,	O
but	O
these	O
are	O
the	O
most	O
recognized	O
,	O
standard	O
designations	O
.	O
</s>
<s>
A	O
divided	B-Algorithm
difference	I-Algorithm
,	O
however	O
,	O
does	O
require	O
further	O
elucidation	O
,	O
as	O
it	O
equals	O
the	O
average	O
derivative	B-Algorithm
between	O
and	O
including	O
LB	O
and	O
UB	O
:	O
</s>
<s>
More	O
formally	O
,	O
Pã	O
is	O
found	O
in	O
the	O
mean	B-Algorithm
value	I-Algorithm
theorem	I-Algorithm
of	O
calculus	O
,	O
which	O
says	O
:	O
</s>
<s>
which	O
links	O
the	O
mean	O
value	O
result	O
with	O
the	O
divided	B-Algorithm
difference	I-Algorithm
:	O
</s>
<s>
As	O
there	O
is	O
,	O
by	O
its	O
very	O
definition	O
,	O
a	O
tangible	O
difference	O
between	O
LB/P0	O
and	O
UB/Pń	O
,	O
the	O
Leibniz	O
and	O
derivative	B-Algorithm
expressions	O
do	O
require	O
divarication	O
of	O
the	O
function	O
argument	O
.	O
</s>
<s>
The	O
quintessential	O
application	O
of	O
the	O
divided	B-Algorithm
difference	I-Algorithm
is	O
in	O
the	O
presentation	O
of	O
the	O
definite	O
integral	O
,	O
which	O
is	O
nothing	O
more	O
than	O
a	O
finite	B-Algorithm
difference	I-Algorithm
:	O
</s>
<s>
Given	O
that	O
the	O
mean	O
value	O
,	O
derivative	B-Algorithm
expression	O
form	O
provides	O
all	O
of	O
the	O
same	O
information	O
as	O
the	O
classical	O
integral	O
notation	O
,	O
the	O
mean	O
value	O
form	O
may	O
be	O
the	O
preferable	O
expression	O
,	O
such	O
as	O
in	O
writing	O
venues	O
that	O
only	O
support/accept	O
standard	O
ASCII	B-Protocol
text	I-Protocol
,	O
or	O
in	O
cases	O
that	O
only	O
require	O
the	O
average	O
derivative	B-Algorithm
(	O
such	O
as	O
when	O
finding	O
the	O
average	O
radius	O
in	O
an	O
elliptic	O
integral	O
)	O
.	O
</s>
<s>
0	O
and	O
either	O
or	O
as	O
boundaries	O
,	O
with	O
the	O
same	O
divided	B-Algorithm
difference	I-Algorithm
found	O
as	O
that	O
with	O
boundaries	O
of	O
0	O
and	O
(	O
thus	O
requiring	O
less	O
averaging	O
effort	O
)	O
:	O
</s>
