<s>
In	O
calculus	O
,	O
the	O
second	B-Algorithm
derivative	I-Algorithm
,	O
or	O
the	O
second-order	O
derivative	B-Algorithm
,	O
of	O
a	O
function	O
is	O
the	O
derivative	B-Algorithm
of	O
the	O
derivative	B-Algorithm
of	O
.	O
</s>
<s>
Roughly	O
speaking	O
,	O
the	O
second	B-Algorithm
derivative	I-Algorithm
measures	O
how	O
the	O
rate	O
of	O
change	O
of	O
a	O
quantity	O
is	O
itself	O
changing	O
;	O
for	O
example	O
,	O
the	O
second	B-Algorithm
derivative	I-Algorithm
of	O
the	O
position	O
of	O
an	O
object	O
with	O
respect	O
to	O
time	O
is	O
the	O
instantaneous	O
acceleration	O
of	O
the	O
object	O
,	O
or	O
the	O
rate	O
at	O
which	O
the	O
velocity	O
of	O
the	O
object	O
is	O
changing	O
with	O
respect	O
to	O
time	O
.	O
</s>
<s>
The	O
last	O
expression	O
is	O
the	O
second	B-Algorithm
derivative	I-Algorithm
of	O
position	O
(	O
x	O
)	O
with	O
respect	O
to	O
time	O
.	O
</s>
<s>
On	O
the	O
graph	B-Application
of	I-Application
a	I-Application
function	I-Application
,	O
the	O
second	B-Algorithm
derivative	I-Algorithm
corresponds	O
to	O
the	O
curvature	O
or	O
concavity	O
of	O
the	O
graph	O
.	O
</s>
<s>
The	O
graph	B-Application
of	I-Application
a	I-Application
function	I-Application
with	O
a	O
positive	O
second	B-Algorithm
derivative	I-Algorithm
is	O
upwardly	O
concave	O
,	O
while	O
the	O
graph	B-Application
of	I-Application
a	I-Application
function	I-Application
with	O
a	O
negative	O
second	B-Algorithm
derivative	I-Algorithm
curves	O
in	O
the	O
opposite	O
way	O
.	O
</s>
<s>
The	O
power	O
rule	O
for	O
the	O
first	B-Algorithm
derivative	I-Algorithm
,	O
if	O
applied	O
twice	O
,	O
will	O
produce	O
the	O
second	B-Algorithm
derivative	I-Algorithm
power	O
rule	O
as	O
follows	O
:	O
</s>
<s>
The	O
second	B-Algorithm
derivative	I-Algorithm
of	O
a	O
function	O
is	O
usually	O
denoted	O
.	O
</s>
<s>
As	O
the	O
previous	O
section	O
notes	O
,	O
the	O
standard	O
Leibniz	O
notation	O
for	O
the	O
second	B-Algorithm
derivative	I-Algorithm
is	O
.	O
</s>
<s>
However	O
,	O
this	O
limitation	O
can	O
be	O
remedied	O
by	O
using	O
an	O
alternative	O
formula	O
for	O
the	O
second	B-Algorithm
derivative	I-Algorithm
.	O
</s>
<s>
This	O
one	O
is	O
derived	O
from	O
applying	O
the	O
quotient	O
rule	O
to	O
the	O
first	B-Algorithm
derivative	I-Algorithm
.	O
</s>
<s>
When	O
written	O
this	O
way	O
(	O
and	O
taking	O
into	O
account	O
the	O
meaning	O
of	O
the	O
notation	O
given	O
above	O
)	O
,	O
the	O
terms	O
of	O
the	O
second	B-Algorithm
derivative	I-Algorithm
can	O
be	O
freely	O
manipulated	O
as	O
any	O
other	O
algebraic	O
term	O
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
inverse	O
function	O
formula	O
for	O
the	O
second	B-Algorithm
derivative	I-Algorithm
can	O
be	O
deduced	O
from	O
algebraic	O
manipulations	O
of	O
the	O
above	O
formula	O
,	O
as	O
well	O
as	O
the	O
chain	O
rule	O
for	O
the	O
second	B-Algorithm
derivative	I-Algorithm
.	O
</s>
<s>
The	O
second	B-Algorithm
derivative	I-Algorithm
of	O
a	O
function	O
can	O
be	O
used	O
to	O
determine	O
the	O
concavity	O
of	O
the	O
graph	O
of	O
.	O
</s>
<s>
A	O
function	O
whose	O
second	B-Algorithm
derivative	I-Algorithm
is	O
positive	O
will	O
be	O
concave	O
up	O
(	O
also	O
referred	O
to	O
as	O
convex	O
)	O
,	O
meaning	O
that	O
the	O
tangent	O
line	O
will	O
lie	O
below	O
the	O
graph	O
of	O
the	O
function	O
.	O
</s>
<s>
Similarly	O
,	O
a	O
function	O
whose	O
second	B-Algorithm
derivative	I-Algorithm
is	O
negative	O
will	O
be	O
concave	O
down	O
(	O
also	O
simply	O
called	O
concave	O
)	O
,	O
and	O
its	O
tangent	O
lines	O
will	O
lie	O
above	O
the	O
graph	O
of	O
the	O
function	O
.	O
</s>
<s>
If	O
the	O
second	B-Algorithm
derivative	I-Algorithm
of	O
a	O
function	O
changes	O
sign	O
,	O
the	O
graph	O
of	O
the	O
function	O
will	O
switch	O
from	O
concave	O
down	O
to	O
concave	O
up	O
,	O
or	O
vice	O
versa	O
.	O
</s>
<s>
Assuming	O
the	O
second	B-Algorithm
derivative	I-Algorithm
is	O
continuous	O
,	O
it	O
must	O
take	O
a	O
value	O
of	O
zero	O
at	O
any	O
inflection	O
point	O
,	O
although	O
not	O
every	O
point	O
where	O
the	O
second	B-Algorithm
derivative	I-Algorithm
is	O
zero	O
is	O
necessarily	O
a	O
point	O
of	O
inflection	O
.	O
</s>
<s>
The	O
relation	O
between	O
the	O
second	B-Algorithm
derivative	I-Algorithm
and	O
the	O
graph	O
can	O
be	O
used	O
to	O
test	O
whether	O
a	O
stationary	O
point	O
for	O
a	O
function	O
(	O
i.e.	O
,	O
a	O
point	O
where	O
)	O
is	O
a	O
local	O
maximum	O
or	O
a	O
local	O
minimum	O
.	O
</s>
<s>
If	O
,	O
the	O
second	B-Algorithm
derivative	I-Algorithm
test	O
says	O
nothing	O
about	O
the	O
point	O
,	O
a	O
possible	O
inflection	O
point	O
.	O
</s>
<s>
The	O
reason	O
the	O
second	B-Algorithm
derivative	I-Algorithm
produces	O
these	O
results	O
can	O
be	O
seen	O
by	O
way	O
of	O
a	O
real-world	O
analogy	O
.	O
</s>
<s>
It	O
is	O
possible	O
to	O
write	O
a	O
single	O
limit	B-Algorithm
for	O
the	O
second	B-Algorithm
derivative	I-Algorithm
:	O
</s>
<s>
The	O
limit	B-Algorithm
is	O
called	O
the	O
second	O
symmetric	B-Algorithm
derivative	B-Algorithm
.	O
</s>
<s>
Note	O
that	O
the	O
second	O
symmetric	B-Algorithm
derivative	B-Algorithm
may	O
exist	O
even	O
when	O
the	O
(	O
usual	O
)	O
second	B-Algorithm
derivative	I-Algorithm
does	O
not	O
.	O
</s>
<s>
The	O
expression	O
on	O
the	O
right	O
can	O
be	O
written	O
as	O
a	O
difference	B-Algorithm
quotient	I-Algorithm
of	O
difference	B-Algorithm
quotients	I-Algorithm
:	O
</s>
<s>
This	O
limit	B-Algorithm
can	O
be	O
viewed	O
as	O
a	O
continuous	O
version	O
of	O
the	O
second	O
difference	O
for	O
sequences	O
.	O
</s>
<s>
However	O
,	O
the	O
existence	O
of	O
the	O
above	O
limit	B-Algorithm
does	O
not	O
mean	O
that	O
the	O
function	O
has	O
a	O
second	B-Algorithm
derivative	I-Algorithm
.	O
</s>
<s>
The	O
limit	B-Algorithm
above	O
just	O
gives	O
a	O
possibility	O
for	O
calculating	O
the	O
second	B-Algorithm
derivative	I-Algorithm
—	O
but	O
does	O
not	O
provide	O
a	O
definition	O
.	O
</s>
<s>
The	O
sign	O
function	O
is	O
not	O
continuous	O
at	O
zero	O
,	O
and	O
therefore	O
the	O
second	B-Algorithm
derivative	I-Algorithm
for	O
does	O
not	O
exist	O
.	O
</s>
<s>
But	O
the	O
above	O
limit	B-Algorithm
exists	O
for	O
:	O
</s>
<s>
Just	O
as	O
the	O
first	B-Algorithm
derivative	I-Algorithm
is	O
related	O
to	O
linear	B-Algorithm
approximations	I-Algorithm
,	O
the	O
second	B-Algorithm
derivative	I-Algorithm
is	O
related	O
to	O
the	O
best	O
quadratic	O
approximation	O
for	O
a	O
function	O
.	O
</s>
<s>
This	O
is	O
the	O
quadratic	O
function	O
whose	O
first	O
and	O
second	B-Algorithm
derivatives	I-Algorithm
are	O
the	O
same	O
as	O
those	O
of	O
at	O
a	O
given	O
point	O
.	O
</s>
<s>
For	O
many	O
combinations	O
of	O
boundary	O
conditions	O
explicit	O
formulas	O
for	O
eigenvalues	B-Algorithm
and	I-Algorithm
eigenvectors	I-Algorithm
of	I-Algorithm
the	I-Algorithm
second	I-Algorithm
derivative	I-Algorithm
can	O
be	O
obtained	O
.	O
</s>
<s>
For	O
example	O
,	O
assuming	O
and	O
homogeneous	O
Dirichlet	O
boundary	O
conditions	O
(	O
i.e.	O
,	O
)	O
,	O
the	O
eigenvalues	O
are	O
and	O
the	O
corresponding	O
eigenvectors	O
(	O
also	O
called	O
eigenfunctions	B-Algorithm
)	O
are	O
.	O
</s>
<s>
For	O
other	O
well-known	O
cases	O
,	O
see	O
Eigenvalues	B-Algorithm
and	I-Algorithm
eigenvectors	I-Algorithm
of	I-Algorithm
the	I-Algorithm
second	I-Algorithm
derivative	I-Algorithm
.	O
</s>
<s>
The	O
second	B-Algorithm
derivative	I-Algorithm
generalizes	O
to	O
higher	O
dimensions	O
through	O
the	O
notion	O
of	O
second	O
partial	O
derivatives	B-Algorithm
.	O
</s>
<s>
If	O
the	O
function	O
's	O
image	O
and	O
domain	O
both	O
have	O
a	O
potential	O
,	O
then	O
these	O
fit	O
together	O
into	O
a	O
symmetric	B-Algorithm
matrix	I-Algorithm
known	O
as	O
the	O
Hessian	O
.	O
</s>
<s>
The	O
eigenvalues	O
of	O
this	O
matrix	O
can	O
be	O
used	O
to	O
implement	O
a	O
multivariable	O
analogue	O
of	O
the	O
second	B-Algorithm
derivative	I-Algorithm
test	O
.	O
</s>
<s>
(	O
See	O
also	O
the	O
second	O
partial	O
derivative	B-Algorithm
test	O
.	O
)	O
</s>
<s>
Another	O
common	O
generalization	O
of	O
the	O
second	B-Algorithm
derivative	I-Algorithm
is	O
the	O
Laplacian	O
.	O
</s>
<s>
The	O
Laplacian	O
of	O
a	O
function	O
is	O
equal	O
to	O
the	O
divergence	B-Application
of	O
the	O
gradient	O
,	O
and	O
the	O
trace	O
of	O
the	O
Hessian	O
matrix	O
.	O
</s>
