Definition:Total Derivative

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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:

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

where $\dfrac {\partial f} {\partial x_k}$ 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:

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


Also see