If we take for example, the problem of $$\int e^x \sin x \quad dx$$
We use the integration by parts technique:
$$\int uv' = uv - \int vu'$$
Setting
$\begin{array}{l l} u = \sin x & \frac{dv}{\color{red}{dx}} = e^x\\ \frac{du}{\color{red}{dx}} = \cos x & v = e^x\\ \end{array}$
However, I've seen people break up the derivative notation by bringing the denominator $dx$ over to the the RHS:
$\begin{array}{l l} u = \sin x & dv = e^x \; \color{red}{dx}\\ du = \cos x \; \color{red}{dx} & v = e^x\\ \end{array}$
I don't understand why people do this. With integration by substitution, this manipulation of the derivative notation is justified. But here it serves no purpose.