# Partial derivative

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of the first order of a function in several variables

The derivative of the function with respect to one of the variables, keeping the remaining variables fixed. For example, if a function $f ( x _ {1}, \dots, x _ {n} )$ is defined in some neighbourhood of a point $( x _ {1} ^ {(0)}, \dots, x _ {n} ^ {(0)} )$, then the partial derivative $( \partial f / \partial x _ {1} ) ( x _ {1} ^ {(0)}, \dots, x _ {n} ^ {(0)} )$ of $f$ with respect to the variable $x _ {1}$ at that point is equal to the ordinary derivative $( d f /d x _ {1} ) ( x _ {1} , x _ {2} ^ {(0)}, \dots, x _ {n} ^ {(0)} )$ at the point $x _ {1} ^ {( 0)}$ of the function $f ( x _ {1} , x _ {2} ^ {(0)}, \dots, x _ {n} ^ {(0)} )$ in the single variable $x _ {1}$. In other words,

$$\left . \frac{\partial f }{\partial x _ {1} } ( x _ {1} ^ {(0)}, \dots, x _ {n} ^ {(0)} ) = \frac{d f }{d x _ {1} } ( x _ {1} , x _ {2} ^ {(0)} \dots x _ {n} ^ {(0)} ) \right | _ {x _ {1} = x _ {1} ^ {(0)} } =$$

$$= \ \lim\limits _ {\Delta x _ {1} \rightarrow 0 } \frac{\Delta _ {x _ {1} } f ( x _ {1} ^ {(0)}, \dots, x _ {n} ^ {(0)} ) }{\Delta x _ {1} } ,$$

where

$$\Delta _ {x _ {1} } f ( x _ {1} ^ {(0)} , \dots, x _ {n} ^ {(0)} ) =$$

$$= \ f ( x _ {1} ^ {(0)} + \Delta x _ {1} , x _ {2} ^ {(0)}, \dots, x _ {n} ^ {(0)} ) - f ( x _ {1} ^ {(0)}, \dots, x _ {n} ^ {(0)} ) .$$

The partial derivatives

$$\tag{* } \frac{\partial ^ {m} f }{\partial x _ {1} ^ {m _ {1} } \cdots \partial x _ {n} ^ {m _ {n} } } ,\ \ m _ {1} + \cdots + m _ {n} = m ,$$

of order $m > 1$ are defined by induction: If the partial derivative

$$\frac{\partial ^ {k- 1} f }{\partial x _ {1} ^ {k _ {1} } \cdots \partial x _ {i} ^ {k _ {i} } \cdots \partial x _ {n} ^ {k _ {n} } } ,\ \ k _ {1} + \cdots + k _ {n} = k - 1 ,$$

has been defined, then by definition

$$\frac{\partial ^ {k} f }{\partial x _ {1} ^ {k _ {1} } \cdots \partial x _ {i} ^ {k _ {i} + 1 } \cdots \partial x _ {n} ^ {k _ {n} } } =$$

$$= \ \frac \partial {\partial x _ {i} } \left ( \frac{\partial ^ {k- 1} f }{\partial x _ {1} ^ {k _ {1} } \cdots \partial x _ {i} ^ {k _ {i} } \cdots \partial x _ {n} ^ {k _ {n} } } \right ) .$$

The partial derivative (*) is also denoted by $D _ {m _ {1} \dots m _ {n} } ^ {m} f$. A partial derivative (*) in which at least two distinct indices $m _ {i}$ are non-zero is called a mixed partial derivative; otherwise, that is, if the partial derivative has the form $\partial ^ {m} f / \partial x _ {i} ^ {m}$, it is called unmixed. Under fairly broad conditions, mixed partial derivatives do not depend on the order of differentiation with respect to the different variables. This holds, for example, if all the partial derivatives under consideration are continuous.

If in the definition of a partial derivative the usual notion of a derivative is replaced by that of a generalized derivative in some sense or another, then the definition of a generalized partial derivative is obtained.