Total derivative

Function

 definition $\dfrac{{\mathrm d}}{{\mathrm d}t}:\left(X_1\times\cdots \times X_n\times{\mathbb R}\to{\mathbb R}\right)\to \left(\left({\mathbb R}\to X_1\times\cdots \times X_n\right)\times {\mathbb R}\right)\to {\mathbb R}$ let $\diamond\ f(x^1,\dots,x^n,t)$ definition $\left(\dfrac{{\mathrm d}}{{\mathrm d}t}f\right)\left(\langle t\mapsto\langle r^1(t),\dots,r^n(t)\rangle,t\rangle\right):=\sum_{j=1}^n \dfrac{\partial f}{\partial x^j}(\langle r^1(t),\dots,r^n(t),t\rangle)\cdot\dfrac{\partial r^j}{\partial t}(t)+\dfrac{\partial f}{\partial t}(\langle r^1(t),\dots,r^n(t),t\rangle)$

Example

The total derivative of the (not explicitly time dependent) function

$f:{\mathbb R}^3\to{\mathbb R}, \ \ \ f(x,y,t):=x^2\cos(y)$

along the trajectory

$R:{\mathbb R}\to{\mathbb R}^2, \ \ \ R:=t\mapsto\langle 7t,-3t^5\rangle$

is

$\left(\dfrac{{\mathrm d}}{{\mathrm d}t}f\right)(R,t)=2\,R_x(t)\cos(R_y(t))\cdot R_x'(t)+R_x(t)^2\sin(R_y(t))\cdot R_y'(t)$

$=2\cdot 7^2\,t\cos(-3t^5)-3\cdot 5\cdot 7^2\,t^{2+4}\sin(-3t^5)$

Reference

Wikipedia: Total derivative