In mathematics, the total derivative of a function f at a point is the best linear approximation near this point of the function with respect to its arguments. Unlike partial derivatives, the total derivative approximates the function with respect to all of its arguments, not just a single one. In many situations, this is the same as considering all partial derivatives simultaneously. The term "total derivative" is primarily used when f is a function of several variables, because when f is a function of a single variable, the total derivative is the same as the ordinary derivative of the function.

Let ${\displaystyle U\subseteq \mathbb {R} ^{n}}$ be an open subset. Then a function ${\displaystyle f:U\to \mathbb {R} ^{m}}$ is said to be (totally) differentiable at a point ${\displaystyle a\in U}$ if there exists a linear transformation ${\displaystyle df_{a}:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}$ such that

The linear map ${\displaystyle df_{a}}$ is called the (total) derivative or (total) differential of ${\displaystyle f}$ at ${\displaystyle a}$. Other notations for the total derivative include ${\displaystyle D_{a}f}$ and ${\displaystyle Df(a)}$. A function is (totally) differentiable if its total derivative exists at every point in its domain.

Conceptually, the definition of the total derivative expresses the idea that ${\displaystyle df_{a}}$ is the best linear approximation to ${\displaystyle f}$ at the point ${\displaystyle a}$. This can be made precise by quantifying the error in the linear approximation determined by ${\displaystyle df_{a}}$. To do so, write

where ${\displaystyle \varepsilon (h)}$ equals the error in the approximation. To say that the derivative of ${\displaystyle f}$ at ${\displaystyle a}$ is ${\displaystyle df_{a}}$ is equivalent to the statement

where ${\displaystyle o}$ is little-o notation and indicates that ${\displaystyle \varepsilon (h)}$ is much smaller than ${\displaystyle \lVert h\rVert }$ as ${\displaystyle h\to 0}$. The total derivative ${\displaystyle df_{a}}$ is the unique linear transformation for which the error term is this small, and this is the sense in which it is the best linear approximation to ${\displaystyle f}$.

The function ${\displaystyle f}$ is differentiable if and only if each of its components ${\displaystyle f_{i}\colon U\to \mathbb {R} }$ is differentiable, so when studying total derivatives, it is often possible to work one coordinate at a time in the codomain. However, the same is not true of the coordinates in the domain. It is true that if ${\displaystyle f}$ is differentiable at ${\displaystyle a}$, then each partial derivative ${\displaystyle \partial f/\partial x_{i}}$ exists at ${\displaystyle a}$. The converse does not hold: it can happen that all of the partial derivatives of ${\displaystyle f}$ at ${\displaystyle a}$ exist, but ${\displaystyle f}$ is not differentiable at ${\displaystyle a}$. This means that the function is very "rough" at ${\displaystyle a}$, to such an extreme that its behavior cannot be adequately described by its behavior in the coordinate directions. When ${\displaystyle f}$ is not so rough, this cannot happen. More precisely, if all the partial derivatives of ${\displaystyle f}$ at ${\displaystyle a}$ exist and are continuous in a neighborhood of ${\displaystyle a}$, then ${\displaystyle f}$ is differentiable at ${\displaystyle a}$. When this happens, then in addition, the total derivative of ${\displaystyle f}$ is the linear transformation corresponding to the Jacobian matrix of partial derivatives at that point.

When the function under consideration is real-valued, the total derivative can be recast using differential forms. For example, suppose that ${\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} }$ is a differentiable function of variables ${\displaystyle x_{1},\ldots ,x_{n}}$. The total derivative of ${\displaystyle f}$ at ${\displaystyle a}$ may be written in terms of its Jacobian matrix, which in this instance is a row matrix: