Definition:Total Derivative

Let $$f \left({x_1, x_2, \ldots, x_n}\right)$$ be a continuous function of multiple variables.

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

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


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

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:


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