im late to the party, BUT....
Ok, assuming the measuring device was absolutely accurate (which they arent) and given that it was your first month of weight loss, then YES, you may have lost only 5 pounds of fat, however,the other 5 pounds lost was NOT MUSCLE. That is a lie. You CANNOT atrophy your muscles that quickly..... the other 5 pounds were merely WATER weight coming off, as you entered into a new caloric deficit for your body... water loss and the glycogen storage assoiated with that as you depleted what you had stored (cuz you were using them, and not replacing them~ the deficit)
LIke on the Biggest loser? The first week or 2--> BIG LOSSES?? 90% of it is water weight loss. But 5 actual pounds of fat in a month is fantastic, plus those 5 pounds of water DO make you smaller? KWM? Water has volume, and inches, etc..... You didnt lose any muscle mass sweetie, so dont freak out
That said, an after workout protein drink is a great idea, especially following weightlifting, or if you need to supplement your protein intake. Whein a caloric deficit (losing fat) it IS a good idea to eat more protein than you might during maintenance..... Im not saying you have to do a high protein/low carb approach, but highER protein will help to preserve muscle mass. Overtime, during your journey, you can/might lose SOME muscle... but not 5 pounds of it, and in ONE MONTH?
Get a protein supplement that fits within your dietary needs....a 100% whey isolate or something similar, that wont have all the extra fat and carbs, if that is something you are tracking and trying to limit.
A Whey protein isolate will digest and be absorbed/utilized very quickly in your body (hence, it makes a great post-workout supplement) and a Casein protein will digest and be absorbed slowly, which keeps you feeling *full* longer. MOST people supplement with whey dominant proteins.
Bottome line... youre doing great, dont buy anythgin from them, and keep up the good work!