Overview
The total derivative is an abstract definition of the defintion which encapsulates the normal single variable derivative definition but can be applied in much broader contexts.
Definition
A multi-variable function
{% f(x_1,x_2,...,x_n) %}
is said to differentiable if there exists a
linear transformation
{% T_a : \mathbb{R}^n \rightarrow \mathbb{R} %}
(called the total derivative, or just derivative)
and a scalar function
{% E(a,v) %} such that
{% f(\vec{a} + \vec{v}) = f(\vec{a}) + T_a(\vec{v}) + |\vec{v}| \times E(\vec{a},\vec{v}) %}
where
{% E(\vec{a},\vec{v}) \rightarrow 0 %}
as
{% |\vec{v}| \rightarrow 0 %}
Note, in this definition, we taken the set of inputs to the function as a single input, a vector {% \vec{v} %}. The definition above then requires that the vector be an element in a Banach space.
Jacobian
The total derivative, represented as a matrix, is called the Jacobian. The Jacobian also features prominantly in integration.
Manifold Definition
When the function in question is defined on a manifold structure, whose range is the reals, {% \mathbb{R} %} the total derivative is defined to have the tangent space as the domain.
A function {% f : U \rightarrow \mathbb{R} %} is differentiable at {% p %} if there is a linear functional {% df_p :T_pU \rightarrow \mathbb{R} %} such that
{% f(p + \vec{v}) - f(p) = df_p (\vec{v}) + r(\vec{v}) %}
where
{% \lim_{\vec{v} \to 0} \frac{r(\vec{v})}{|\vec{v}|} = 0 %}
and {% T_pU %} is the tangent plane at {% p %}.
(see Doria pg. 5)
Note that within the context of manifold theory, the total derivative is written using the language of differential forms.