The directional derivative is a vector gives the rate of change of a function at a given point.
Definition
Let be a unit vector in the x-y plane. The rate of change of at a point in the direction is the directional derivative, written as any of the following:
This gives the gradient of the surface of at the point , in the direction
Info
The del operator (or nabla symbol) , in differential calculus, gives the gradient vector of a function:
Vector Properties
Since the directional derivative is essentially the dot product of two vectors, we can rewrite it as:
(Since the magnitude of a unit vector is 1)
Hence, the gradient, is a maximum when , i.e. . Similarly, it is the lowest when , i.e. .
This gives a geometrical interpretation of
• the direction of is the direction of steepest ascent of . • the length of , is the slope of the surface in the direction of steepest ascent. • the direction of is the direction of steepest descent of f . • is perpendicular to the level curves of f .
Examples
1: Simple directional derivative
Find the directional derivative of at in the direction from towards
First, we need to find the directional unit vector, .
Now we need to find the gradient of f, at :
Therefore, the directional derivative is: