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 $x\mapsto f(x_{1},x_{2},\dots ,x_{i-1},a_{i},x_{i+1},\dots ,x_{n})$ 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.