In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are useful in vector calculus and differential geometry.
The partial derivative of a function with respect to the variable is written as . The partial-derivative symbol is a rounded letter, distinguished from the straight d of total-derivative notation. The notation was introduced by Adrien-Marie Legendre and gained general acceptance after its reintroduction by Carl Gustav Jacob Jacobi.
Suppose that is a function of more than one variable. For instance,
A graph of
. We want to find the partial derivative at
constant; the corresponding tangent line is parallel to the
It is difficult to describe the derivative of such a function, as there are an infinite number of tangent lines to every point on this surface. Partial differentiation is the act of choosing one of these lines and finding its slope. Usually, the lines of most interest are those that are parallel to the -axis, and those that are parallel to the -axis.
This is a slice of the graph at the right at
A good way to find these parallel lines is to treat the other variable as a constant. For example, to find the tangent line of the above function at that is parallel to the -axis, we treat as a constant one. The graph and this plane are shown on the right. On the left, we see the way the function looks on the plane . By finding the tangent line on this graph, we discover that the slope of the tangent line of at that is parallel to the -axis is three. We write this in notation as
or as "The partial derivative of with respect to at is 3."
The function can be reinterpreted as a family of functions of one variable indexed by the other variables:
In other words, every value of defines a function, denoted , which is a function of one real number. That is,
Once a value of is chosen, then determines a function which sends to :
In this expression, is a constant, not a variable, so is a function of only one real variable, that being . Consequently the definition of the derivative for a function of one variable applies:
The above procedure can be performed for any choice of . Assembling the derivatives together into a function gives a function which describes the variation of in the direction:
This is the partial derivative of with respect to . Here is a rounded called the partial derivative symbol. To distinguish it from the letter , is sometimes pronounced "der", "del", "dah", or "partial" instead of "dee".
In general, the partial derivative of a function in the direction at the point is defined to be:
In the above difference quotient, all the variables except are held fixed. That choice of fixed values determines a function of one variable , and by definition,
In other words, the different choices of index a family of one-variable functions just as in the example above. This expression also shows that the computation of partial derivatives reduces to the computation of one-variable derivatives.
An important example of a function of several variables is the case of a scalar-valued function on a domain in Euclidean space (e.g. on or ). In this case has a partial derivative with respect to each variable . At the point , these partial derivatives define the vector
This vector is called the gradient of at . If is differentiable at every point in some domain, then the gradient is a vector-valued function which takes the point to the vector . Consequently the gradient determines a vector field.
The volume of a cone depends on height and radius
Consider the volume of a cone; it depends on the cone's height and its radius according to the formula
The partial derivative of with respect to is
It describes the rate with which a cone's volume changes if its radius is varied and its height is kept constant. The partial derivative with respect to is
and represents the rate with which the volume changes if its height is varied and its radius is kept constant.
Now consider by contrast the total derivative of with respect to and . They are, respectively
We see that the difference between the total and partial derivative is the elimination of indirect dependencies between variables in the latter.
Now suppose that, for some reason, the cone's proportions have to stay the same, and the height and radius are in a fixed ratio :
This gives the total derivative:
Equations involving an unknown function's partial derivatives are called partial differential equations and are common in physics, engineering, and other sciences and applied disciplines.
For the following examples, let be a function in .
First-order partial derivatives:
Second-order partial derivatives:
Second-order mixed derivatives:
Higher-order partial and mixed derivatives:
When dealing with functions of multiple variables, some of these variables may be related to each other, and it may be necessary to specify explicitly which variables are being held constant. In fields such as statistical mechanics, the partial derivative of with respect to , holding and constant, is often expressed as
Formal definition and properties
Like ordinary derivatives, the partial derivative is defined as a limit. Let be an open subset of and a function. We define the partial derivative of at the point with respect to the -th variable as
Even if all partial derivatives exist at a given point , the function need not be continuous there. However, if all partial derivatives exist in a neighborhood of and are continuous there, then is totally differentiable in that neighborhood and the total derivative is continuous. In this case, we say that is a function. We can use this fact to generalize for vector valued functions () by carefully using a componentwise argument.
The partial derivative can be seen as another function defined on and can again be partially differentiated. If all mixed second order partial derivatives are continuous at a point (or on a set), we call a function at that point (or on that set); in this case, the partial derivatives can be exchanged by Clairaut's theorem:
- ↑ This can also be expressed as the adjointness between the product space and function space constructions.