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)	$\underset{x\to x_0}{lim}\frac{f(x,y_0)-f(x_0,y_0)}{x-x_0}$ $\underset{h\to 0}{lim}\frac{f(x_0+h,y_0)-f(x_0,y_0)}{h}$	$\underset{y\to y_0}{lim}\frac{f(x_0,y)-f(x_0,y_0)}{y-y_0}$ $\underset{h\to 0}{lim}\frac{f(x_0,y_0+h)-f(x_0,y_0)}{h}$
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	$\frac{d}{d{x}_i}f(a_1,a_2,\dots ,a_{i-1},x_i,a_{i+1},\dots ,a_n){|}_{x_i=a_i}$. 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)	$\underset{x_i\to a_i}{lim}\frac{f(a_1,a_2,\dots ,a_{i-1},x_i,a_{i+1},\dots ,a_n)-f(a_1,a_2,\dots ,a_n)}{x_i-a_i}$
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.