Infinitesimal time intervals use I've a question, that maybe will sound obvious, on the use of infinitesimal quantities.
Consider the expression for the acceleration in non inertial frames. 
$\frac{d\vec{v}}{dt}=\frac{d\vec{v'}}{dt}+2\vec{\Omega}\times\vec{v'}+\frac{d\vec{\Omega}}{dt}\times\vec{r'}+\vec{\Omega}\times(\vec{\Omega}\times\vec{r'})$
The expression is not important (is just used as an example). Here both $\vec{\Omega}$ and $\frac{d\vec{\Omega}}{dt}$ appear (a vector and its derivative). Now here we are trying to find the variation of $\vec{v}$ in a infinitesimal interval $dt$. Nevertheless when we write $\vec{\Omega}$ we mean $\vec{\Omega}(t)$, i.e. the angular velocity at the time istant $t$. How can we talk about the time interval $dt$ considering $\vec{\Omega}(t)$ (at a particular istant $t$)? Is this justified by the fact that $dt$ is an infinitesimal quantity?
 A: Derivative is defined at a point not for an infinitesimal interval.
$$\text {If $y=f(x)$} \Longrightarrow \; y’(x)|_{x=a}=\frac {\mathrm dy}{\mathrm dx}\bigg|_{x=a}=f’(a)=\lim_{\Delta x\to 0}{\frac {f(a+\Delta x)-f(a)}{\Delta x}}$$
$\large{\frac {\mathrm d\vec \Omega}{\mathrm dt}}$ doesn’t represent a fraction. That represents derivative of the $\vec \Omega$ relative to $t$ at a time instant like $t$ or $t_1$ or etc. In other words, when we write $\frac {\mathrm d\vec \Omega}{\mathrm dt}$, this means $\frac{\mathrm d\vec \Omega}{\mathrm dt}\bigg|_{t=t}$ and this doesn’t mean “variation of $\vec \Omega$” per “variation of $t$”. This just means derivative of $\vec \Omega$ relative to $t$.

Now here we are trying to find the variation of $\vec v$ in an infinitesimal interval $\mathrm dt$. 

We don’t need to find the variation of the $\vec v$ in an infinitesimal interval $dt$. We need to find derivative of the $\vec v$ relative to $t$ at each time instant $t$. Acceleration vector of a particle at each time instant is defined by derivative of the velocity vector of that particle at that time instant not anything else.
A: In order for the derivative to exist, the function must be well defined on an interval which includes that instant. Therefore the derivative is also defined on an interval arbout that instant.
