nLab total derivative

Idea

Given a function $f$ of $n$ variables, and given $n$ functions $\{g_i\}$ of one variable, then the total derivative of the composite function $t \mapsto f(g_1(t),\cdots, g_n(t))$ is (if it exists) simply its derivative with respect to $t$, but understood as a linear combinationof the partial derivatives of $f$, via the chain rule:

$$\frac{d f}{d t} = \sum_i \frac{\partial f}{\partial g_i} \, \frac{d g_i}{d t} \,.$$

This has various evident generalizations. One is the horizontal derivative in variational calculus, see at variational bicomplex.