James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 14.6 Directional Derivatives and the Gradient Vector 947
sects S in a curve C. (See Figure 3.) The slope of the tangent line T to C at the point P is
the rate of change of z in the direction of u.
z
T
P(x¸, y¸, z¸)
TEC Visual 14.6A animates Figure 3
by rotating u and therefore T.
S
Q(x, y, z)
C
Pª (x ¸, y ¸, 0)
ha
h
hb
u
Qª (x, y , 0 )
y
FIGURE 3
x
If Qsx, y, zd is another point on C and P9, Q9 are the projections of P, Q onto the
xy-plane, then the vector P9Q9 B is parallel to u and so
P9Q9 B − hu − kha, hb l
for some scalar h. Therefore x 2 x 0 − ha, y 2 y 0 − hb, so x − x 0 1 ha, y − y 0 1 hb,
and
Dz
h − z 2 z 0
h
− f sx 0 1 ha, y 0 1 hbd 2 f sx 0 , y 0 d
h
If we take the limit as h l 0, we obtain the rate of change of z (with respect to distance)
in the direction of u, which is called the directional derivative of f in the direction of u.
2 Definition The directional derivative of f at sx 0 , y 0 d in the direction of a
unit vector u − ka, bl is
D u f sx 0 , y 0 d − lim
h l 0
f sx 0 1 ha, y 0 1 hbd 2 f sx 0 , y 0 d
h
if this limit exists.
By comparing Definition 2 with Equations 1, we see that if u − i − k1, 0 l, then
D i f − f x and if u − j − k0, 1 l, then D j f − f y . In other words, the partial derivatives of f
with respect to x and y are just special cases of the directional derivative.
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