Directional derivative and gradient examples
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:
(x,y) | = 2xy | (x,y) | = x2 | ||||
(3, 2) | = 12 | (3, 2) | = 9 |
| ∇f(3, 2) = 12i + 9j = (12, 9). |
(b) Let u = u1i + u2j be a unit vector. The directional derivative at (3,2) in the direction of u is
| Duf(3, 2) | = ∇f(3, 2) ⋅ u | ||
| = (12i + 9j) ⋅ (u1i + u2j) | |||
| = 12u1 + 9u2. | (1) |
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 = = = = (1∕ , 2∕ ). |
| 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 = = (2∕ , 1∕ ). |
| 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 | ||
= + = = 15, |
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).)
(
(
(3
(3
=
=
= (1
)
+
= 
= (2
)
+
= 
+
=
= 15