
418 probability density function
d
––
dx
COUNT
; M
ONTE
C
ARLO METHOD
; M
ONTY
H
ALL PROB
-
LEM
;
ODDS
;
RANDOM WALK
;
STATISTICS
;
TWO
-
CARD
PUZZLE
.
probability density function See
DISTRIBUTION
.
Proclus See E
UCLID
’
S POSTULATES
.
product rule The
DERIVATIVE
of the product of two
functions fand gis given by the product rule:
For example, we have (xsinx) = 1 .sinx+ x.cosx=
sinx+ xcosx. The rule can be proved using the limit
definition of the derivative as follows:
Alternatively, one can recognize that the quantity
f(x+ h) · g(x+ h) is the formula for the area of a rectan-
gle, one that contains the smaller rectangle of area f(x) ·
g(x).Writing a formula for the area of the L-shaped
region between the two rectangles, dividing by h, and
taking the limit as hbecomes small leads to the same
formula for the product rule.
The product rule can be generalized to apply to
any finite product of functions. For example, for the
product of three functions we have:
(f(x) · g(x) · h(x))′= f′(x) · g(x) · h(x) + f(x) · g′(x) · h(x)
+ f(x) · g(x) · h′(x)
If one of the functions in a product is a constant k,
then the product rule shows:
(kf(x))′= 0 · f(x) + k· f′(x) = kf′(x)
Two applications of the product rule give the second
derivative of a product of two functions:
(f(x) · g(x))′′ = f(x) · g′′(x) + 2f′(x) · g′(x) + f′′(x) · g(x)
In general, the nth derivative of a product of two
functions is a sum of products containing
BINOMIAL
COEFFICIENT
s:
This result is called Leibniz’s theorem.
See also
CHAIN RULE
;
HIGHER DERIVATIVE
;
QUO
-
TIENT RULE
.
projection Any mapping of a geometric figure onto a
PLANE
to produce a two-dimensional image of that fig-
ure is called a projection. For instance, the daytime
shadow cast by an outdoor object is an example of a
projection onto the ground. Since the Sun is a great dis-
tance from the Earth, rays of sunlight are essentially
PARALLEL
, and shadows cast by it are parallel projec-
tions. Shadows cast by a single point of light, however,
such as the flame of a candle, have different shapes
than those cast by the sun. Such projections are called
central projections.
French mathematician and engineer G
IRARD
D
ESARGUES
(1591–1661) observed that the central pro-
jection of any
CONIC SECTION
is another conic section.
(For instance, the shape cast on the ground by the cir-
cular rim of a flashlight is an
ELLIPSE
.) This led him to
study those properties of geometric figures that remain
unchanged under central projections, thereby founding
the field of
PROJECTIVE GEOMETRY
.
In
VECTOR
analysis, the projection of a vector a
onto a vector bis a vector parallel to bwhose length is
the length of the “shadow” cast by aif the two vectors
are placed at the same location in space and the “rays
of light” casting the shadow are parallel and
PERPEN
-
DICULAR
to b. Thus the projection of aonto bis a vec-
tor of the form xb, for some value x, with the property
that the vector connecting the tip of xbto the tip of ais
perpendicular to b. Using the
DOT PRODUCT
of vectors,
this yields the equation:
fx gx n
kfxg x
nknk
k
n
() () () ()
() () ( )
⋅
()
=⎛
⎝
⎜⎞
⎠
⎟⋅−
=
∑
0
d
dx fx gx fx h gx h fx gx
h
fx h gx h fx gx h fx gx h fx gx
h
fx h fx
hgx
h
h
h
() () lim ()()()()
lim ()()()()()()()()
lim ()()
(
⋅= +⋅ +− ⋅
=+⋅ +− ⋅ ++ ⋅ +− ⋅
=+− ⋅+
()
→
→
→
0
0
0hhfx
gx h gx
h
fxgx fxgx
)()
()()
() () () ()
+⋅
+−
=′⋅+⋅
′
⎛
⎝
⎜⎞
⎠
⎟
d
dx fg df
dx gf
dg
dx
⋅
()
=⋅+⋅