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:


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

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


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