Definition:Total Derivative

Definition
Let $f \left({x_1, x_2, \ldots, x_n}\right)$ 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:


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

where $\displaystyle \frac {\partial f} {\partial x_k}$ 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:


 * $\displaystyle \frac {\mathrm d f}{\mathrm d x_k} = \frac {\partial f} {\partial x_1} \frac {\mathrm d x_1}{\mathrm d x_k} + \frac {\partial f} {\partial x_2} \frac {\mathrm d x_2}{\mathrm d x_k} + \cdots + \frac {\partial f} {\partial x_k} + \cdots + \frac {\partial f} {\partial x_n} \frac {\mathrm d x_n}{\mathrm d x_k}$