For a function f:U→**R** (U⊆**R**^{n} some open set) and a point **x**=(x_{1},...,x_{n}), define

f_{i,x}(t) =
f(x_{1},...,x_{i-1},t,x_{i+1},...,x_{n}).

Then f

_{i,x} is a function of one

variable when t is near x

_{i}, and we can do

calculus with it. The

*partial derivative* of f with respect to x

_{i} is defined to be

∂f/∂x_{i} = df_{i,x}/dt.

Note the unfortunate double usage of **x** and especially x_{i} to denote both a vector or a scalar variable *and* a particular value of that variable. Unfortunately, this practice is so common that I feel I must abide by it.
This is a function ∂f/∂x

_{i} : U→R.
If the derivative on the

RHS does not exist, the partial derivative on the

LHS doesn't, either.

By tradition, the same letters are re-used, instead of new ones like t. Your multivariable calculus textbook may well follow this "convention", and give a seemingly different definition. The only purpose of this is to make a confusing subject more confusing.
Another notation for ∂f/∂x_{i} is f_{xi}. This notation is particularly useful for taking multiple partial derivatives: for a function f(x,y),

f_{xx} = ∂^{2}f/∂x^{2} =
∂/∂x(∂f/∂x);

f_{xy} = ∂^{2}f/∂y∂x =
∂/∂y(∂f/∂x);

f_{yx} = ∂^{2}f/∂x∂y =
∂/∂x(∂f/∂y);

f_{yy} = ∂^{2}f/∂y^{2} =
∂/∂y(∂f/∂y).

In particular, the

Laplacian of f is Δf =
f

_{xx}+f

_{yy}.

*WHEN* f is differentiable, it turns out that

∇f = (∂f/∂x_{1},...,∂f/∂x_{n}).

But even if all partial derivatives

exist at

**x**, f might not be differentiable there --

be careful!

Contrary to what you might expect (or even read about in various places, once upon a time even on E2...), it is not necessarily true that ∂^{2}f/∂x_{i}∂x_{j} = ∂^{2}f/∂x_{j}∂x_{i}. You *cannot*, in general, interchange the order of the derivatives.