D
irectional
D
erivatives
Up to this point we have discussed how to take the derivative in the
x
-direction or in
the
y
-direction. These were the partial derivatives with respect to
x
and
y
, respectively.
But what happens if we want to take the derivative in a direction that is not parallel to the
coordinate axes?
It seems reasonable to suspect that if we were to go halfway in between the
x
and
y
axes, the directional derivative would be the average of the partial derivatives with
respect to
x
and
y
. But how do we combine them?
Recall that when we discussed the interpretation of the derivative, it was the change in
the function if the input were to change by one unit. So, we want to define the directional
derivative as the change in the function for a unit step in that particular direction.
We shall specify the direction in which we wish to take the derivative as
u
=
u
1
i
+
u
2
j
.
Because we want to take a unit step in this direction, we require that
u
be a unit vector. If
we are given a vector that is not a unit vector, we need to convert it to a unit vector first
by using the fact that
v
u
v
. With that, we are ready to define the direction derivative.
Directional Derivative of
f
(
x
,
y
) at (
a
,
b
) in the Direction of a Unit Vector u
If
u
=
u
1
i
+
u
2
j
is a unit vector, we define the direction derivative
f
u
at the point
(
a
,
b
) by
12
0
Rate of change of
( , ) in the
(,
)
(
,
(,)
l
im
direction of
at the point ( , )
h
fxy
)
f
ah
ubh
u
fa
b
fab
ab
h
u
u
provided that the limit exists.
Notice that if
u
=
i
, so
u
1
= 1 and
u
2
= 0, then the directional derivative is
f
x
(
a
,
b
) since
0
)
(
,
)
(,) l
x
h
fa hb
f ab
h
i
. Similarly, if
u
=
j
, then the direction
derivative is equal to
f
y
(
a
,
b
).