The important thing to bear in mind with children is that all are different. Not just yours, your relatives and friends- around the globe, every child is different, unique. While your daughter or others may have adapted within a short space or cope with change well, that's not to say that all are the same in this respect.
Without going into a long, winding novella- understand that some children just plain don't like change- unlike adults, they aren't used to or even remotely capable of grasping at such an age the cognative function to process "I need to give this a chance"- they know what they lie and don't like-if they don't get it, well- you already know the response!
My son when he started daycare had been spoiled for attention and had myself and his mother around constantly for the first 9 months until she was back at work, then it was "taking turns" as we worked opposite shifts- but the point being, he was always used to the same faces, this is what he was used to. Remember, children aren't you and I, while it won't bother you in the slightest to go to a shopping center, supermarket or walk down a street and see other people, these little ones aren't, so it's a shock to their system as even if you have family around, these become familiar and let's face it, the world is far from identical from one day to the next.
The adaptation from having mostly her around was thankfully easy, he didn't mind at all. However when he started to go to daycare, this was a different matter. It took him about 3 months to finally get used to going and not have a good cry for about an hour every time.
We started with a "toes first" approach of just one day a fortnight, then once a week which is all that we thankfully needed- but it still took those three months.
It's a special kind of feeling you get when you see your child upset, especially when you feel you're responsible. Thankfully, most daycares will have some advice or guidance you can take to make the change easier for the child to adapt- from a routine to follow down to getting them familiar before even attending by socialising with unfamiliar faces to get them used to seeing other people.
A lot of it comes down to you, though- some parents take the "bandaid" approach of getting their chil(ren) to daycare, giving them a kiss on the head and a goodby and leaving, while others will loiter for a time and let their child go off to do their own thing and leave then.
This is different for everyone and each child- daycare staff are all trained and used to different children, they know what they're doing and your child is safe with them, so calm yourself down, don't get anxious- let your child feel that this is all ok and normal.
Help them to feel more excited, find out how they've done during the day and what they've enjoyed doing- is it a particular activity? Playing with a particular other child or children? Perhaps a toy? Maybe even a particular staff member? Find something they like and use that as something to refer to so they get positive vibes as the prospect- rather than them seeing you get in a state because of "what's coming"- kids aren't stupid or soft, they do pick up on this.
Daycare is an important part of development as unless you've already been taking your child to any groups or activities with others to help socialize them, this may well be the first time they've had to face not just strangers, but other children, the concept of sharing, being away from you- it's a big thing, but they look to you for support, approval, safety....everything- if they see you upset, this isn't good.
Speak to the staff, possibly see if they'd mind you checking in at some point in the day if you feel it's necessary. Remember though it's all aiming to the end goal of them being used to and accepting that they'll be at daycare for a period of time, letting you do what you need to, whatever that may be. Dropping in frequently and making it a habit will make the transition worse, prologing the entire process.
Believe me when I say that this is an ongoing thing as there's still preschool, high school, university and them moving out, getting married...chances are you'll be the one that needs to do the adapting, not them, as it doesn't get much easier as they get older!