Definition:Total Derivative

Definition
Let $\map f {x_1, x_2, \ldots, x_n}$ be a continuous real function of multiple variables.

Let each of $x_1, x_2, \ldots, x_n$ be continuous real functions of a single independent variable $t$.

Then the total derivative of $f$ with respect to $t$ is defined as:


 * $\ds \frac {\d f}{\d t} = \sum_{k \mathop = 1}^n \frac {\partial f} {\partial x_k} \frac {\d x_k}{\d t} = \frac {\partial f} {\partial x_1} \frac {\d x_1}{\d t} + \frac {\partial f} {\partial x_2} \frac {\d x_2}{\d t} + \cdots + \frac {\partial f} {\partial x_n} \frac {\d x_n}{\d t}$

where $\dfrac {\partial f} {\partial x_k}$ is the partial derivative of $f$ with respect to $x_k$.

Note that in the above definition, nothing precludes $t$ from being one of the instances of $x_k$ itself.

So we have that the total derivative of $f$ with respect to $x_k$ is defined as:


 * $\dfrac {\d f} {\d x_k} = \dfrac {\partial f} {\partial x_1} \dfrac {\d x_1}{\d x_k} + \dfrac {\partial f} {\partial x_2} \dfrac {\d x_2} {\d x_k} + \cdots + \dfrac {\partial f} {\partial x_k} + \cdots + \dfrac {\partial f} {\partial x_n} \dfrac {\d x_n}{\d x_k}$

Also see

 * Chain Rule for Real-Valued Functions