properties of differential

[smallabc]

Hi, are the following steps valid?

Case 1 :

P = (1+t)S

lnP = ln(1+t) + lnS　( taking natural log )

dlnP = dln(1+t) + dlnS　( where d stands for small differential )

Case 2 :

P = (1+t)S

dP = d[(1+t)S]　( where d stands for small differential )

dP = d(1+t)dS