Partial 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 more than one variable, where
is one of the input variables to
. Fix a choice
and fix the values of all the other variables. The partial derivative of
with respect to
at the point, denoted
, or
, is defined as the derivative at
of the function that sends
to
at
for the same fixed choice of the other input variables.
For a function of two variables
Suppose is a real-valued function of two variables
, i.e., the domain of
is a subset of
. Suppose
is a point in the domain of
, i.e., it's the point with
and
(here,
are actual numerical values). We define the partial derivatives at
as follows:
Item | For partial derivative with respect to ![]() |
For partial derivative with respect to ![]() |
---|---|---|
Notation | ![]() Also denoted ![]() ![]() |
![]() Also denoted ![]() ![]() |
Definition as derivative | ![]() ![]() ![]() |
![]() ![]() ![]() |
Definition as limit (using derivative as limit of difference quotient) | ![]() ![]() |
![]() ![]() |
Definition as directional derivative | Directional derivative at ![]() ![]() |
Directional derivative at ![]() ![]() |
For a function of multiple variables
The notation here gets a little messy, so read it carefully. We consider a function of
variables, which we generically denote
respectively. Consider a point
in the domain of the function. In other words, this is a point where
.
Suppose is a natural number in the set
.
Item | Value for partial derivative with respect to ![]() |
---|---|
Notation | ![]() Also denoted ![]() ![]() |
Definition as derivative | ![]() ![]() ![]() ![]() |
Definition as a limit (using derivative as limit of difference quotient) | ![]() |
Definition as a directional derivative | Directional derivative in the positive ![]() |
Definition as a function
Generic definition
Suppose is a function of more than one variable, where
is one of the input variables to
. The partial derivative of
with respect to
, denoted
, or
is defined as the function that sends points in the domain of
(including values of all the variables) to the partial derivative with respect to
of
(i.e., the derivative treating the other inputs as constants for the computation of the derivative). In particular, the domain of the partial derivative of
with respect to
is a subset of the domain of
.
We can compute the partial derivative of relative to each of the inputs to
.
MORE ON THE WAY THIS DEFINITION OR FACT IS PRESENTED: We first present the version that deals with a specific point (typically with asubscript) in the domain of the relevant functions, and then discuss the version that deals with a point that is free to move in the domain, by dropping the subscript. Why do we do this?
The purpose of the specific point version is to emphasize that the point is fixed for the duration of the definition, i.e., it does not move around while we are defining the construct or applying the fact. However, the definition or fact applies not just for a single point but for all points satisfying certain criteria, and thus we can get further interesting perspectives on it by varying the point we are considering. This is the purpose of the second, generic point version.
For a function of two variables
Suppose is a real-valued function of two variables
, i.e., the domain of
is a subset of
. The partial derivatives of
with respect to
and
are both functions of two variables each of which has domain a subset of the domain of
.
Item | For partial derivative with respect to ![]() |
For partial derivative with respect to ![]() |
---|---|---|
Notation | ![]() Also denoted ![]() ![]() |
![]() Also denoted ![]() ![]() |
Definition as derivative | It is the derivative of the function ![]() ![]() |
It is the derivative of the function ![]() ![]() |
Definition as limit (using derivative as limit of difference quotient) | ![]() |
![]() |
Definition as directional derivative | Directional derivative with respect to a unit vector in the positive ![]() |
Directional derivative with respect to a unit vector in the positive ![]() |
For a function of multiple variables
Item | Value for partial derivative with respect to ![]() |
---|---|
Notation | ![]() Also denoted ![]() ![]() |
Definition as derivative | It is the derivative of the function ![]() ![]() |
Definition as a limit (using derivative as limit of difference quotient) | ![]() |
Definition as a directional derivative | Directional derivative in the positive ![]() |
Graphical interpretation
For a function of two variables at a point
Suppose is a function of two variables
and
is a point in the domain of the function. Consider the graph of
in three-dimensional space, given by
.
We have the following:
Partial derivative | Graphical interpretation |
---|---|
The partial derivative ![]() ![]() |
The slope of the tangent line at ![]() ![]() ![]() |
The partial derivative ![]() ![]() |
The slope of the tangent line at ![]() ![]() ![]() |
For a function of multiple variables at a point
Suppose is a function of
variables
and suppose
is a point in the domain of
. Consider the graph of
in
given by:
For any , we define the partial derivative
, also denoted
, as follows:
- First, consider the intersection of the graph of
with the plane given by the set of
equations
for all
. This is a plane parallel to the
-plane.
- In this plane, consider the slope of the tangent line at
. This is the value of the partial derivative.
Related notions
Domain considerations
As already noted in the definition of partial derivative, the domain of the partial derivative of a function with respect to a variable is a subset of the domain of the function. However, we can actually say a little more.
For a function of two variables
Suppose is a function of two variables
. Then, a necessary condition for us to make sense of the partial derivative
at a point
is that
be defined on a small open interval about the point
on the line
. Note that it is not necessary that
actually be defined in an open ball surrounding the point
-- the only thing that matters is that
be defined under slight perturbations of
, holding
constant.
Similar remarks apply to : a necessary condition for us to make sense of the partial derivative
at a point
is that
be defined on a small open interval about the point
on the line
.
Consider, for instance, a function defined on the set , i.e., the set
. It makes sense to try computing the partial derivative
at all points in the subset
, i.e., all points whose
-coordinate is strictly between
and
, but the
-coordinate is allowed to take the extreme values 0 and 1. Similarly, it makes sense to try computing the partial derivative
at all points in the subset
, i.e., all points whose
-coordinate is strictly between
and
, but the
-coordinate is allowed to take the extreme values 0 and 1.
Note that the above only refers to the points at which it makes sense to try computing the partial derivative. It may still turn out that the partial derivative does not exist at many of these points.
Caveats
Value of partial derivative depends on all inputs
For further information, refer: Value of partial derivative depends on all inputs
For instance, consider:
Then, we have:
and:
Note that each of the expressions involves both the variables and
. In particular, this means that the value of
at a point depends on both the
-coordinate and the
-coordinate of the point. Thus, for instance:
Despite the same -value of 2 in both cases, the
-values are different because of differences in the input
-values.
Similarly, consider:
Despite the same -value of 4 in both cases, the
-values are different because of differences in the input
-values.
Meaning of partial derivative depends on entire coordinate system
For further information, refer: Meaning of partial derivative depends on entire coordinate system
This is a very subtle but very important point. It says that the partial derivative with respect to one variable depends not only on the choice of that particular variable, but on the choice of the other variables that are being kept constant for the purpose of computing the partial derivative. If a coordinate transformation is performed that changes what those other variables are, that could affect the value of the partial derivative.
This has a very real-world corollary. In economics and social science, we often talk of the partial derivative with respect to one variable as measuring what happens ceteris paribus on the other variables. However, the notion of ceteris paribus on other variables depends on what the other variables are. If we redefine the coordinate system to change that meaning, the partial derivative can change.
Consider the function:
In this case, we have:
Now, suppose we consider in terms of
and
. Then, we have
. Rewriting
in terms of
and
, we get:
In other words, we can define as a function of two variables
and
. If we use the letter
to denote this new function, we get:
In this case, we have:
Note that the two partial derivatives with respect to are not equal. The reason for this is that in the first case, we are taking the partial derivative with respect to
keeping
constant, whereas in the second case, we are taking the partial derivative with respect to
keeping
constant. In this case, when we increase
slightly, the value of
decreases to keep the total constant.
Here's the geometric interpretation:
- In the first case, where we are computing
, we are geometrically computing the directional derivative along the positive
-direction, i.e., along a line with
-coordinate.
- In the second case, where we are computing
, we are geometrically computing (up to scalar multiples) the directional derivative along lines with
constant. These lines are downward sloping with a slope of
.