Partial derivative

Definition at a point

Generic definition

Suppose $f$ is a function of more than one variable, where $x$ is one of the input variables to $f$. Fix a choice $x=x_{0}$ and fix the values of all the other variables. The partial derivative of $f$ with respect to $x$, denoted $\partial f/\partial x$, or $f_{x}$, is defined as the derivative at $x_{0}$ of the function that sends $x$ to $f$ at $x$ for the same fixed choice of the other input variables.

For a function of two variables

Suppose $f$ is a real-valued function of two variables $x,y$, i.e., the domain of $f$ is a subset of $\mathbb {R} ^{2}$. Suppose $(x_{0},y_{0})$ is a point in the domain of $f$. We define the partial derivatives at $(x_{0},y_{0})$ as follows:

Item	For partial derivative with respect to $x$	For partial derivative with respect to $y$
Notation	${\frac {\partial f(x,y)}{\partial x}}|_{(x,y)=(x_{0},y_{0})}$ Also denoted $f_{x}(x_{0},y_{0})$ or $f_{1}(x_{0},y_{0})$	${\frac {\partial f(x,y)}{\partial y}}|_{(x,y)=(x_{0},y_{0})}$ Also denoted $f_{y}(x_{0},y_{0})$ or $f_{2}(x_{0},y_{0})$
Definition as derivative	${\frac {d}{dx}}f(x,y_{0})|_{x=x_{0}}$. In other words, it is the derivative (at $x=x_{0}$) of the function $x\mapsto f(x,y_{0})$	${\frac {d}{dy}}f(x_{0},y)|_{y=y_{0}}$. In other words, it is the derivative (at $y=y_{0}$) of the function $y\mapsto f(x_{0},y)$.
Definition as limit (using derivative as limit of difference quotient)	$\lim _{x\to x_{0}}{\frac {f(x,y_{0})-f(x_{0},y_{0})}{x-x_{0}}}$ $\lim _{h\to 0}{\frac {f(x_{0}+h,y_{0})-f(x_{0},y_{0})}{h}}$	$\lim _{y\to y_{0}}{\frac {f(x_{0},y)-f(x_{0},y_{0})}{y-y_{0}}}$ $\lim _{h\to 0}{\frac {f(x_{0},y_{0}+h)-f(x_{0},y_{0})}{h}}$
Definition as directional derivative	Directional derivative at $(x_{0},y_{0})$ with respect to a unit vector in the positive $x$-direction.	Directional derivative at $(x_{0},y_{0})$ with respect to a unit vector in the positive $y$-direction.

For a function of multiple variables

The notation here gets a little messy, so read it carefully. We consider a function $f$ of $n$ variables, which we generically denote $(x_{1},x_{2},\dots ,x_{n})$ respectively. Consider a point $(a_{1},a_{2},\dots ,a_{n})$ in the domain of the function. In other words, this is a point where $x_{1}=a_{1},x_{2}=a_{2},\dots ,x_{n}=a_{n}$.

Suppose $i$ is a natural number in the set $\{1,2,3,\dots ,n\}$.

Item	Value for partial derivative with respect to $x_{i}$
Notation	${\frac {\partial }{\partial x_{i}}}f(x_{1},x_{2},\dots ,x_{n})|_{(x_{1},x_{2},\dots ,x_{n})=(a_{1},a_{2},\dots ,a_{n})}$ Also denoted $f_{x_{i}}(a_{1},a_{2},\dots ,a_{n})$ or $f_{i}(a_{1},a_{2},\dots ,a_{n})$
Definition as derivative	${\frac {d}{dx_{i}}}f(a_{1},a_{2},\dots ,a_{i-1},x_{i},a_{i+1},\dots ,a_{n})|_{x_{i}=a_{i}}$. In other words, it is the derivative (evaluated at $a_{i}$) of the function $x\mapsto f(a_{1},a_{2},\dots ,a_{i-1},x_{i},a_{i+1},\dots ,a_{n})$ with respect to $x_{i}$, evaluated at the point $x_{i}=a_{i}$.
Definition as a limit (using derivative as limit of difference quotient)	$\lim _{x_{i}\to a_{i}}{\frac {f(a_{1},a_{2},\dots ,a_{i-1},x_{i},a_{i+1},\dots ,a_{n})-f(a_{1},a_{2},\dots ,a_{n})}{x_{i}-a_{i}}}$
Definition as a directional derivative	Directional derivative in the positive $x_{i}$-direction.

Definition as a function

Generic definition

The partial derivative of a function $f$ of $n$ variables with respect to one of its inputs is defined as the function that sends each point to the partial derivative of $f$ with respect to that input at that point. The domain of this is defined as the set of those points in the domain of $f$ where the partial derivative exists. In particular, the domain of the partial derivative of $f$ is a subset of the domain of $f$.

Note a key fact: the general expression for the partial derivative with respect to any of the inputs is an expression in terms of all the inputs. For instance, the general expression for $f_{x}(x,y)$ is an expression involving both $x$ and $y$. This is because, even though the $y$-coordinate is kept constant when calculating the partial derivative at a particular point, that constant value need not be the same at all the points.

MORE ON THE WAY THIS DEFINITION OR FACT IS PRESENTED: We first present the version that deals with a specific point (typically with a $\{\}_{0}$ subscript) 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 $f$ is a real-valued function of two variables $x,y$, i.e., the domain of $f$ is a subset of $\mathbb {R} ^{2}$. The partial derivatives of $f$ with respect to $x$ and $y$ are both functions of two variables each of which has domain a subset of the domain of $f$.

Item	For partial derivative with respect to $x$	For partial derivative with respect to $y$
Notation	${\frac {\partial f(x,y)}{\partial x}}$ Also denoted $f_{x}(x,y)$ or $f_{1}(x,y)$	${\frac {\partial f(x,y)}{\partial y}}$ Also denoted $f_{y}(x,y)$ or $f_{2}(x,y)$
Definition as derivative	It is the derivative of the function $x\mapsto f(x,y)$, treating $y$ as an unknown constant	It is the derivative of the function $y\mapsto f(x,y)$, treating $x$ as an unknown constant
Definition as limit (using derivative as limit of difference quotient)	$\lim _{h\to 0}{\frac {f(x+h,y)-f(x,y)}{h}}$	$\lim _{h\to 0}{\frac {f(x,y+h)-f(x,y)}{h}}$
Definition as directional derivative	Directional derivative with respect to a unit vector in the positive $x$-direction.	Directional derivative with respect to a unit vector in the positive $y$-direction.

For a function of multiple variables

Item	Value for partial derivative with respect to $x_{i}$
Notation	${\frac {\partial }{\partial x_{i}}}f(x_{1},x_{2},\dots ,x_{n})$ Also denoted $f_{x_{i}}(x_{1},x_{2},\dots ,x_{n})$ or $f_{i}(x_{1},x_{2},\dots ,x_{n})$
Definition as derivative	It is the derivative of the function $x_{i}\mapsto f(x_{1},x_{2},\dots ,x_{i-1},x_{i},x_{i+1},\dots ,x_{n})$ with respect to $x_{i}$, where all the other variables are treated as unknown constants while doing the differentiation.
Definition as a limit (using derivative as limit of difference quotient)	$\lim _{h\to 0}{\frac {f(x_{1},x_{2},\dots ,x_{i-1},x_{i}+h,x_{i+1},\dots ,x_{n})-f(x_{1},x_{2},\dots ,x_{n})}{h}}$
Definition as a directional derivative	Directional derivative in the positive $x_{i}$-direction.