Directional derivative
This article describes an analogue for functions of multiple variables of the following term/fact/notion for functions of one variable: derivative
Contents
Definition at a point
Generic definition
Suppose is a function of many variables. Consider the domain of
as a subset of Euclidean space. Fix a direction in this space and a point in the domain. Then, the directional derivative at the point in the direction is the derivative of the function with respect to movement of the point along that direction, at the specific point.
For a function of two variables
Suppose is a function of two variables
. Suppose
is a unit vector (i.e., we have
). Suppose
is a point in the domain of
We define the directional derivative of
at
in the direction of
as follows.
Item | Value |
---|---|
Notation | ![]() ![]() |
Definition as a limit | ![]() |
Definition as an ordinary derivative | ![]() |
For a function of multiple variables
Suppose is a function of variables
. Suppose
is a unit vector (i.e., we have
). Suppose
is a point in the domain of
. The directional derivative of
at
in the direction of
is defined as follows.
Item | Value |
---|---|
Notation | ![]() ![]() |
Definition as a limit | ![]() |
Definition as an ordinary derivative | ![]() |
For a function of multiple variables in vector notation
Suppose is a function of a vector variable
. Suppose
is a unit vector and
is a point in the domain of
. The directional derivative of
at
in the direction of
is denoted and defined as below.
Item | Value |
---|---|
Notation | ![]() ![]() |
Definition as a limit | ![]() |
Definition as an ordinary derivative | ![]() |
Definition as a function
Generic definition
Suppose is a function of many variables. Consider the domain of
as a subset of Euclidean space. Fix a direction in this space. Then, the directional derivative in the direction is the function sending a point in the domain of
to the derivative of the function with respect to movement of the point along that direction.
For a function of two variables
Suppose is a function of two variables
, with domain a subset of
. Suppose
is a unit vector (i.e., we have
). Then, the directional derivative in the direction of
is a function with domain a subset of the domain of
, defined as the function that sends any point in the domain of
to the directional derivative of
in the direction of
at the point.
Item | Value |
---|---|
Notation | ![]() ![]() |
Definition as a limit | ![]() |
Definition as a partial derivative | ![]() ![]() |
For a function of multiple variables
Suppose is a function of variables
. Suppose
is a unit vector (i.e., we have
). We define and denote the directional derivative as below.
Item | Value |
---|---|
Notation | ![]() ![]() |
Definition as a limit | ![]() |
Definition as an ordinary derivative | ![]() ![]() |
For a function of multiple variables in vector notation
Suppose is a function of a vector variable
. Suppose
is a unit vector. We define and denote the directional derivative of
in the direction of
below.
Item | Value |
---|---|
Notation | ![]() ![]() |
Definition as a limit | ![]() |
Definition as an ordinary derivative | ![]() ![]() |
Graphical interpretation
For a function of two variables
For further information, refer: graph of a function of two variables
The directional derivative in the direction of a unit vector
at a point
can be determined as follows: first, intersect the graph of the function with the plane
. This plane is perpendicular to the
-plane and its intersection with the
-plane is the line through
in the direction of the unit vector
.
This intersection can be thought of as the graph of a function of one variable, where the point is treated as the origin, the direction
is the independent variable axis, and the
-axis direction is the dependent variable axis. Now, the directional derivative is the slope of this graph for dependent variable value of 0.
Relation with gradient vector
Version type | Statement |
---|---|
at a point, in vector notation (multiple variables) | Suppose ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() The right side here is the dot product of vectors. |
generic point, in vector notation (multiple variables) | Suppose ![]() ![]() ![]() ![]() The right side here is a dot product of vectors. The equality holds whenever the right side makes sense. |
generic point, point-free notation (multiple variables) | Suppose ![]() ![]() ![]() ![]() The right side here is a dot product of vector-valued functions (the constant function ![]() ![]() |
Relation with partial derivatives
Partial derivatives as directional derivatives
Version type | Statement |
---|---|
Generic | For a function ![]() ![]() ![]() ![]() ![]() |
Directional derivative in terms of partial derivatives
If the gradient vector at a point exists, then it is a vector whose coordinates are the corresponding partial derivatives of the function. Thus, conditional to the existence of the gradient vector, we have that:
In particular, for a function of two variables and a unit vector
, we have the following, assuming that the gradient vector exists (i.e., that the function is differentiable):