stylistic:style_transfer, linguistic:paraphrasing
Definition

Like ordinary derivatives, the partial derivative is defined as a limit. Let U be an open subset of R n {\displaystyle \mathbb {R} ^{n}} and f: U → R {\displaystyle f:U\to \mathbb {R} } a function. The partial derivative of f at the point a = (a 1, …, a n) ∈ U {\displaystyle \mathbf {a} =(a_{1},\ldots,a_{n})\in U} with respect to the i -th variable x is defined as

∂ ∂ x i f (a) = lim h → 0 f (a 1, …, a i − 1, a i + h, a i + 1 …, a n) − f (a 1, …, a i, …, a n) h = lim h → 0 f (a + h e i) − f (a) h. {\displaystyle {\begin{aligned}{\frac {\partial }{\partial x_{i}}}f(\mathbf {a})&=\lim _{h\to 0}{\frac {f(a_{1},\ldots,a_{i-1},a_{i}+h,a_{i+1}\,\ldots,a_{n})\ -f(a_{1},\ldots,a_{i},\dots,a_{n})}{h}}\\&=\lim _{h\to 0}{\frac {f(\mathbf {a} +h\mathbf {e_{i}})-f(\mathbf {a})}{h}}\,.\end{aligned}}}

Where e i {\displaystyle \mathbf {e_{i}} } is the unit vector of i -th variable x. Even if all partial derivatives ∂ f / ∂ x i (a) {\displaystyle \partial f/\partial x_{i}(a)} 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 C function. This can be used to generalize for vector valued functions, f: U → R m {\displaystyle f:U\to \mathbb {R} ^{m}}, by carefully using a componentwise argument.

The partial derivative ∂ f ∂ x {\textstyle {\frac {\partial f}{\partial x}}} can be seen as another function defined on U and can again be partially differentiated. If the direction of derivative is not repeated, it is called a mixed partial derivative. If all mixed second order partial derivatives are continuous at a point (or on a set), f is termed a C function at that point (or on that set); in this case, the partial derivatives can be exchanged by Clairaut's theorem:

∂ 2 f ∂ x i ∂ x j = ∂ 2 f ∂ x j ∂ x i. {\displaystyle {\frac {\partial ^{2}f}{\partial x_{i}\partial x_{j}}}={\frac {\partial ^{2}f}{\partial x_{j}\partial x_{i}}}.}