Qualitative analysis is used to identify elements by looking at a change in their color, physical state, solubility and other visible properties when specific reagents are added to the chemicals being analyzed. There are a large number of reagents that are used in this process and each element forms a unique product that can be identified visibly.
To find the elements present in the powders contained in the bottle, they would first have to be converted into an aqueous solution. In qualitative analysis, cations are divided into 6 different groups and anions into 3 groups.
The number of elements that lie in each group is quite large and it would not be possible for me to list all of them here. You can check for the same at the link provided below.
Cations of the various groups are identified using reagents like hydrochloric acid, compounds that can provide a sulfide ion, carbonic acid, Nessler's reagent, etc. Individual elements form different kinds of compounds with the reagents that behave differently, for example some dissolve in basic solutions while others dissolve in acidic solutions. The list of reagents and reactions is a large one as each element behaves differently.
Cations too can be detected in ways similar to that of anions. Please look at the details of the reactions in the link provided below.