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Concept
Partial derivative
Summary
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 used in vector calculus and differential geometry.
The partial derivative of a function with respect to the variable is variously denoted by
It can be thought of as the rate of change of the function in the -direction.
Sometimes, for , the partial derivative of with respect to is denoted as Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly signified by the notation, such as in:
The symbol used to denote partial derivatives is ∂. One of the first known uses of this symbol in mathematics is by Marquis de Condorcet from 1770, who used it for partial differences. The modern partial derivative notation was created by Adrien-Marie Legendre (1786), although he later abandoned it; Carl Gustav Jacob Jacobi reintroduced the symbol in 1841.
Like ordinary derivatives, the partial derivative is defined as a limit. Let U be an open subset of and a function. The partial derivative of f at the point with respect to the i-th variable xi is defined as
Even if all partial derivatives exist at a given point a, the function need not be continuous there. However, if all partial derivatives exist in a neighborhood of a and are continuous there, then f is totally differentiable in that neighborhood and the total derivative is continuous. In this case, it is said that f is a C1 function. This can be used to generalize for vector valued functions, , by carefully using a componentwise argument.
The partial derivative can be seen as another function defined on U and can again be partially differentiated. If the direction of derivative is repeated, it is called a mixed partial derivative.
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