The
derivative
of f at P
o
(x
o
,y
o
) in the direction of the unit
vector u=u
1
i+u
2
j
is the number
df
ds
u,Po
=
0
lim
s
12
(
,
)
(
,
)
o
o
o
o
f x
su
y
f x
y
s
provided the
limit exits.
The
gradient vector (gradient)
of f(x,y) at a point P
o
(x
o
,y
o
) is
the vector
f
dx
dy
ij
obtained by evaluating the
partial derivatives of f at P
o
.
If the partial derivatives of f(x,y) are defined at P
o
(x
o
,y
o
), then
,
(
)
,
o
p
u Po
f
u
the scalar product of the
gradient f a P
o
and
u.
Properties of the Directional Derivative
D
u
f=
cos
ff
u
1.
The function f increases most rapidly when cos
=1,
or when
u
is the direction of
f.
That is, at each
point P in its domain, f increases most rapidly in the
direction of the gradient vector
f
at P.
The
derivatives in this direction is
D
u
f =
cos 0
.
2.
Similarly, f decreases most rapidly in the direction of

f.
The derivative in this direction is
D
u
f =
(
cos( ))
.
3.
Any direction
u
orthogonal to the gradient is a
direction of a zero change in f because
then equals
/2
and
D
u
f=
/ 2
0
0
At every point (x
o
,y
o
) in the domain f(x,y), the gradient of f is
normal to the level curve through (x
o
,y
o
).
The
tangent plane
at the point P
o
(x
o
,y
o
,z
o
) on the level surface
f(x,y,z)=c is the plane through P
o
normal to
o
P
f
.
The
normal line
of the surface at P
o
is the line through P
o
is
the line through P
o
parallel to
o
p
f
.