For a differentiable, multivariate function
Developing the formula
The rate of change of
in the direction of the unit vector , moving from position Partial derivatives are special cases of directional derivatives along x- or y-axis. So directional derivative can also be rewritten as:
or dot product of gradient of
and unit vector:
Formula
The rate of change of a differentiable, multivariate function
in the direction of unit vector : In words, the directional derivative is the dot product of gradient and unit vector, both of which can be represented by algebraic vectors of components.
Process
- Convert vector to obtain a unit vector
- Find the partial derivative at the given point to arrive at the gradient of the function
- Plug in the directional derivative formula
Convert vector
Intuitively, if
However, since a unit vector has a magnitude/length of 1, we should normalize the vector components by dividing them by the magnitude:
Sometimes the direction of changing
Examples
(Workbook)
An example
Let
and . Find the directional derivative of at (3,4) in the direction of Confirm
is a unit vector Check if its magnitude = 1
Find the gradient of
at : Plug in the directional derivative formula
Attention
The directional derivative of a function in the direction of steepest ascent is the gradient normalized into unit vector.
Find the directional derivative of
at in the direction of steepest ascent. Find the gradient at
Therefore,
Find the unit vector
Since the direction of steepest ascent is the gradient, we just need to convert the gradient into unit vector. Compute the gradient’s magnitude:
The corresponding unit vector:
Find the directional derivative
We observe that the directional derivative in the direction of the steepest ascent is the same as the gradient after being converted into a unit vector.