Total derivative

of a composite function

The derivative with respect to $t$ of the function $y=f(t,u_1,\dotsc,u_m)$ which depends on the variable $t$ not only directly but also via the intermediate variables $u_1=u_1(t,x_1,\dotsc,x_n),\dotsc,u_m=u_m(t,x_1,\dotsc,x_n)$. It is calculated by the formula

$$\frac{dy}{dt}=\frac{\partial f}{\partial t}+\frac{\partial f}{\partial u_1}\frac{\partial u_1}{\partial t}+\dotsb+\frac{\partial f}{\partial u_m}\frac{\partial u_m}{\partial t},$$

where $\partial f/\partial t$, $\partial f/\partial u_1,\dots,\partial f/\partial u_m$, $\partial u_1/\partial t,\dots,\partial u_m/\partial t$ are partial derivatives (cf. Partial derivative).