Example 1
Let
f (x, y) = x2y. (a) Find
f (3, 2). (b)
Find the derivative of f in the direction of (1,2) at the point
(3,2).
Solution: (a) The gradient is just the vector of partial derivatives. The partial derivatives of f at the point (x, y) = (3, 2) are:
| = 2xy | = x2 | |||
| = 12 | = 9 |
(b) Let u = u1i + u2j be a unit vector. The directional derivative at (3,2) in the direction of u is
To find the directional derivative in the direction of the vector (1,2), we need to find a unit vector in the direction of the vector (1,2). We simply divide by the magnitude of (1, 2).
u = = = |
| Duf (3, 2) | = 12u1 +9u2 | |
| = |
Example 2
For the f of Example 1, find the directional derivative of f at the point (3,2) in the direciton of (2, 1).
Solution: The unit vector in the direction of (2, 1) is
| u = |
| Duf (3, 2) | = 12u1 +9u2 | |
| = |
Example 3
For the f of Example 1 at the point (3,2), (a) in which direction is the directional derivative maximal, (b) what is the directional derivative in that direction?
Solution: (a) The gradient points in the direction of the maximal
directional derivative. The directional derivative is maximal in the
direction of (12,9). (A unit vector in that direction is
u = (12, 9)/
= (4/5, 3/5).)
(b) The magnitude of the gradient is this maximal directional
derivative, which is
||(12, 9)|| =
= 15. Hence the
directional derivative at the point (3,2) in the direction of (12,9)
is 15.
We could double-check by calculating the result using equation (1) and the unit vector u = (4/5, 3/5). Then we find that
| Duf (3, 2) | = 12u1 +9u2 | |
| = |
Example 4
For the f of Example 1 at the point (3,2), (a) what is the directional
derivative in the direction (-3,4) (which is perpendicular to
f (3, 2)), and (b) what is the directional derivative in the direction
(-4,-3) (which is opposite of the direction of
f (3, 2))?
Solution: (a) The directional derivative must be zero. (b)
The directional derivative must be
- ||
f (3, 2)||, which is
-15. (You can verify these by calculating the results directly
using equation (1).)