Total differentials

Total differentials are useful but how do you technically go from them to differentials of functions. Let’s define $f(x)$ as a continuous function.


$$ df= \frac{\partial f}{\partial x} dx,$$

where df is called the total differential. If x(t) is a function t then how do we get this explicit dependence?

$$ f = \int df = \int  \frac{\partial f}{\partial x} dx, $$

where we have used the fundamental theorem of calculus. Now we can perform a change of variables using the total differential of x ie. $dx=dx/dt dt$:

$$ f = \int  \frac{\partial f}{\partial x} dx =  \int  \frac{\partial f}{\partial x} \frac{dx}{dt} dt,$$

which can be differentiated with respect to time to give

$$ \frac{df}{dt} =\frac{\partial f}{\partial x} \frac{dx}{dt}. $$

Here we changed the variables of the integrals (so be careful of the integral limits)