When I first learnt differentiation, I was taught that $\frac{dy}{dx}$ is not a fraction and is, in fact, the limit of a fraction; I was told that, for example, I wasn't allowed to conclude that $dy=dxf(x)$ from $\frac{dy}{dx}=f(x)$.
Instead, I should write:
$\delta{y} \approx {\delta{x}f(x)} \quad\to \quad\frac{\delta{y}}{\delta{x}} \approx{f(x)} \quad\to\quad\lim_{x \to \infty}(\frac{\delta{y}}{\delta{x}})=\frac{dy}{dx} $
However, I've seen manipulations such as:
$d\textbf{F}=dq\times(\textbf{v}\times\textbf{B})\quad\to\quad d\textbf{F}=(Idt)(\textbf{v}\times\textbf{B})=I(\textbf{v}dt\times\textbf{B})\quad\to\quad I(d \textbf{l} \times\textbf{B}) $
or
$dq=nA\textbf{s}e \quad \to \quad \frac{dq}{dt}=\frac{nA\textbf{s}e}{dt}=nA\textbf{v}e$
or
$d\textbf{s}=r\times d\textbf{$\theta $} \quad \to \quad \frac{d\textbf{s}}{dt}=r\times \frac{d\theta}{dt} \quad \to \quad \textbf{v}=r\times\omega\\$
Are such calculations mathematically correct?
Or should the last example be written in the following way:
$\\s=r\theta \quad \to \quad \frac{ds}{dt}=\frac{d}{dt}(r\theta)=r\frac{d\theta}{dt}+\theta \frac{dr}{dt} \quad \to \quad since\quad\frac{dr}{dt}=0 \quad \to \quad v=r\times \frac{d\theta}{dt}\\$
If the above version of the derivation of the angular velocity is the correct one, what would the correct version of the first derivation be?
