Let
u
= <
a, b
>
be the unit
vector
(a vector of length one) on the
xy
plane which
indicates the direction we are moving. Then we define the following:
Definition of the Directional Derivative
The directional derivative of a function
z = f
(
x, y
) in the direction of the unit vector
u
= <
a, b
>
,
denoted by
)
,
(
y
x
f
D
u
, is defined the be the following:
b
y
x
f
a
y
x
f
y
x
f
D
y
x
)
,
(
)
,
(
)
,
(
+
=
u
Notes
1. Geometrically, the directional derivative is used to calculate the slope of the surface
z = f
(
x, y
). That is, to calculate the slope of the surface at the point
)
,
,
(
0
0
0
z
y
x
,
where
)
,
(
0
0
0
y
x
f
z
=
, we compute the following:
b
y
x
f
a
y
x
f
y
x
f
D
b
a
vector
unit
of
direction
in
z
y
x
po
at
Surface
of
Slope
y
x
)
,
(
)
,
(
)
,
(
.
)
,
,
(
int
0
0
0
0
0
0
0
0
0
+
=
=
= <
u
u
2.
The vector
u
= <
a, b
>
must
be
a
unit
vector
. If we want to compute the directional
derivative of a function in the direction of the vector
v
and
v
is not a unit vector, we
compute
v
v

v

v
u


1
=
=
.
3.
The direction of the unit vector
u
can be expressed in terms of the angle
θ
between
the vector
u
and the
x
axis. In this case,
= <
sin
,
cos
u
(note,
u
is a unit vector
since
1
1
sin
cos


2
2
=
=
+
=
u
) and the directional derivative can be expressed
as
sin
)
,
(
cos
)
,
(
)
,
(
y
x
f
y
x
f
y
x
f
D
y
x
+
=
u
.
4.
Computationally, the directional derivative represents the rate of change of the
function
f
in the direction of the unit vector
u
.
2