Multivariate Calculus - Bruce E. Shapiro

76 LECTURE 11. GRADIENTS AND THE DIRECTIONAL DERIVATIVE
Returning to equation 11.3, let u = uh/h where h = ‖h‖,i.e., any vector that is
parallel to h but has magnitude u. Then
f(P + (h/u)u) − f(P)
lim
= u · ∇f(P)
h→0 h
u
(11.9)
Since u is finite (but fixed), then h/u → 0 as h → 0, so that
u · ∇f(P) =
f(P + (h/u)u) − f(P)
lim
h/u→0 h/u
(11.10)
Finally, if we define a new k = h/u then we can make the following generalization
of the derivative.
Definition 11.6 The directional derivative of f in the direction of u at P
is given by
f(P + ku) − f(P)
D u f(P) = u · ∇f(P) = lim
(11.11)
k→0 k
Theorem 11.2 The following are equivalent:
1. f(x, y) is differentiable at P
2. f(x, y) is locally linear at P
3. D u f(P) is defined at P
4. The partial derivatives ∂f/∂x and ∂f/∂y exists and are continuous in some
neighborhood of P.
Figure 11.1: Geometric interpretation of partial derivatives as the slopes of a function
in planes parallel to the coordinate planes.
Example 11.3 Find the directional derivative of f(x, y) = 2x 2 + xy − y 2 at p =
(3, −2) in the direction a = i − j.
Revised December 6, 2006. Math 250, Fall 2006