14.6: DIRECTIONAL DERIVATIVES AND THE
GRADIENT VECTOR
KIAM HEONG KWA
1.
Directional Derivatives
The directional derivative
of a function
f
(
x,y
) at (
x
0
,y
0
) in the di
rection of a unit vector
u
=
h
a,b
i
is deﬁned by
D
u
f
(
x
0
,y
0
) = lim
h
→
0
f
(
x
0
+
ha,y
0
+
hb
)

f
(
x
0
,y
0
)
h
or, in vector notation, by
D
u
f
(
x
0
) = lim
h
→
0
f
(
x
0
+
h
u
)

f
(
x
0
)
h
,
where
x
0
=
h
x
0
,y
0
i
, provided the limit exists. It indicates the rate of change
of
f
in the direction of
u
. In particular, note that
D
i
f
(
x
0
,y
0
) =
f
x
(
x
0
,y
0
) and
D
j
f
(
x
0
,y
0
) =
f
y
(
x
0
,y
0
)
.
Theorem 1.
If
f
is diﬀerentiable, then it has a directional derivative
in the direction of
any
unit vector
u
=
h
a,b
i
. Furthermore,
D
u
f
(
x,y
) =
f
x
(
x,y
)
a
+
f
y
(
x,y
)
b.
In vector notation,
(1.1)
D
u
f
(
x
) =
∇
f
(
x
)
·
u
,
where
x
=
h
x,y
i
and
(1.2)
∇
f
(
x
) =
f
x
(
x
)
i
+
f
y
(
x
)
j
,
the latter of which is called the gradient
of
f
and also denoted by
grad
f
.
If the unit vector
u
makes an angle
α
with the positive
x
axis, so
that
u
=
h
cos
α,
sin
α
i
, then
(1.3)
D
u
f
(
x,y
) =
f
x
(
x,y
) cos
α
+
f
y
(
x,y
) sin
α.
Date
: September 26, 2010.
1