What does she lose that is important to her

Lyddie loses a lot of important things in the book. She loses her father, who walks out on the family; she loses her mother, who ends up in an insane asylum and eventually passes away; and she loses the family farm, which is sold off to pay for her mother's care. Later on in the story, she also loses her job at the factory after she's falsely branded a trouble-maker for daring to stand up to sexual harassment. Long before that incident, though, she lost quite a lot of dignity in her job as a general dogsbody at Cutler's Tavern, where she slaved away for sixteen-hours a day for an absolute pittance.
All of these things are important to Lyddie, but what makes her such an inspiring person is that she gains quite a lot during the story, too, such as new friends, a sense of self-worth, and a passion for education and learning. Overall, one would have to say that Lyddie's gained considerably more than she's lost.

Much of this story is a story about Lyddie and loss, so I am unsure of exactly which specific loss the question is asking about. Early in the book, Lyddie's home is attacked by a bear, and Lyddie's mother takes it as a bad sign. She tries taking the entire family to a relative's house, but Lyddie and her brother want nothing to do with it. They want to keep their home up and running in case their father returns home. The two make it through the winter; however, Lyddie and her brother are soon informed that her mom has sold the house. This could be the loss that the question is asking about. She has lost her family's home, and there is nothing she can do about it.  
It's also possible that the question is pointing toward events that happen near the story's conclusion. Lyddie protects a fellow factory girl from the boss's sexual advances, but her actions result in her being fired and black listed. Lyddie has lost her job and her source of income.  

h(t) = log_5(4-t)^2 Find the derivative of the function

Derivative of a function h with respect to t is denoted as h'(t).
 The given function: h(t) = log_5(4-t)^2 is in a form of a logarithmic function.
From the derivative for logarithmic functions, we follow:
 
d/(dx)log_a(u) =((du)/(dx))/(u*ln(a))
 
By comparison: log_5(4-t)^2 vs.log_a(u) we should let:
a=5 and u = (4-t)^2
 For the derivative of u, recall the Chain Rule formula:
d/(dx)(f(g(x)))= f'(g(x))*g'(x)
Using u=(4-t)^2 , we let:
f(t) = t^2
g(t) = 4-t as the inner function
f'(t)= 2t
f'(g(t))= 2*(4-t)
g'(t)= (-1)
Following the Chain Rule formula, we get:
d/(dx) (4-t)^2= 2 *(4-t)*(-1)
d/(dx) (4-t)^2= -2*(4-t)
or
(du)/(dx)=-2*(4-t)
 Plug-in the values:
u =(4-t)^2 ,     a=5  ,  and (du)/(dx)=-2*(4-t)
 in the d/(dx)log_a(u) =((du)/(dx))/(u*ln(a)) , we get:
d/(dx) (log_5(4-t)^2) = ((-2)*(4-t))/((4-t)^2ln(5))
Cancel out common factor (4-t):
d/(dx) (log_5(4-t)^2) = -2/((4-t)ln(5))
 or h'(t)= -2/((4-t)ln(5))

how is macbeth presented as insecure

Macbeth, like Shakespeare's other protagonists, is both an individual and a larger-than-life representative of one or more abstract qualities of human nature. Critics and commentators have often identified "ambition" as Macbeth's chief characteristic. I would take issue with that assessment.
Macbeth is driven by the witches' prophecy, by Lady Macbeth's urging, and finally by the fact that he's gotten in so deep—as he says himself, "so far stepped in blood"—that he has to continue with more killings, as if an uncontrollable chain reaction has been started. His is less a case of ambition than one of "insecurity"—for lack of a more precise term. When the witches are first encountered, Banquo observes Macbeth's frightened reaction to their apparently favorable predictions. Macbeth has perhaps already contemplated murder as a means of advancement but has no confidence in his ability to carry it out, or perhaps he has enough of a conscience to be shaken by the possibility he might do such a thing. But he's not secure enough in his present status as a nobleman and a military leader to resist an evil course that might be suggested to him by Lady Macbeth.
Macbeth's inability to stand up to her prodding is partly due to that insecurity. At the same time, his actions seem almost inexplicable in the context of what we recognize as "normal" human behavior. From our present-day perspective, Macbeth and Lady Macbeth can be seen as sociopaths. The miracle of Shakespeare's dramatic and poetic skill, however, carries us along and engages our sympathy for Macbeth in his tragic fate. When he kills Duncan, Macbeth realizes that he has also destroyed his own life: "Renown and grace is dead.....the wine of life is drawn / And the mere lees is left, this vault to brag of."
When the ghost of Banquo appears, Macbeth becomes hysterical. He is not a coward, but "insecurity" is an apt description of the reason for his behavior, though it's a huge understatement. Macbeth is in the midst of a trance-like dream. It's as if the ground has vanished beneath his feet, and just as in a nightmare, he has no control over himself or the action of the dream. At the end, he straps on his armor to fight Macduff, but less because he wants to defend himself and more because fighting would be no better and no worse than doing anything else. He is trapped in this tale of life "told by an idiot, signifying nothing," as he so eloquently says.
Macbeth is, in fact, a profoundly weak and insecure man, propelling himself into a course of action in which he does not believe and which he knows is wrong. But this is an existential drama in which a man seemingly has no choice but to destroy those around him and himself. We, the audience, identify with him and the story because the play is a stunning parable, presenting a larger-than-life version of the smaller weaknesses and sins to which we all, as insecure human beings, are subject.

Why is the farm important to the story in E. B. White's Charlotte's Web?

The farm is the main setting for the novel Charlotte's Web. It is there that Wilbur faces his trials and is saved by various characters. Interestingly, both of Wilbur's "saviors" are female characters. Wilbur is first saved by the young girl Fern, who sends him to Zuckerman's farm. Wilbur meets a variety of characters at the farm, some helpful and some sneaky. Most notably, the farm is where Wilbur meets his second savior, Charlotte the spider.
Since Wilbur is getting bigger, he will soon be slaughtered for food. Therefore, the farm not only serves as a setting, but as a sort of antagonist for Wilbur. It started out as a happy home, but eventually became almost a prison for Wilbur. At the farm, there is no escape from his ultimate fate. The farm does become safe again when Charlotte steps in with her webs to save Wilbur.  
The farm is important because it is the setting and has a dual symbolism: home and prison.

Rainsford is uncomfortable when he notices that the general is studying him. What is the general doing here?

Rainsford notices Zaroff "studying" him at dinner. The text says that Rainsford is enjoying dinner and finding Zaroff a great host. Rainsford must feel like he has lucked out. He has fallen overboard and found himself on an island that is operated by a fellow gentleman hunter.

He was finding the general a most thoughtful and affable host, a true cosmopolite.

However, even at this early moment, Rainsford has a sense that Zaroff isn't all that he seems to be and that Zaroff hasn't told him everything.

But there was one small trait of the general's that made Rainsford uncomfortable. Whenever he looked up from his plate he found the general studying him, appraising him narrowly.

Rainsford will soon put two and two together and realize exactly why Zaroff is studying him so intently. Zaroff hunts people for fun. He intentionally captures them and makes sure that they are in good health. He does this so that his human prey is as fit and cunning as possible. He wants the challenge. Unfortunately, most of Zaroff's victims do not offer him much of a challenge. He has grown bored with hunting sailors. They are not creative prey.

The hunting was not good last night. The fellow lost his head. He made a straight trail that offered no problems at all. That's the trouble with these sailors; they have dull brains to begin with, and they do not know how to get about in the woods. They do excessively stupid and obvious things. It's most annoying.

Zaroff is studying Rainsford because Zaroff wants to get a good idea of what kind of hunt Rainsford will give him. Zaroff knows that Rainsford is a fit and accomplished hunter, and Zaroff now wants to test his hunting ability against Rainsford's hopeful ability to be a worthy and creative prey.

"My dear fellow," said the general, "have I not told you I always mean what I say about hunting? This is really an inspiration. I drink to a foeman worthy of my steel—at last." The general raised his glass, but Rainsford sat staring at him.
"You'll find this game worth playing," the general said enthusiastically. "Your brain against mine. Your woodcraft against mine. Your strength and stamina against mine."

Calculus of a Single Variable, Chapter 8, 8.2, Section 8.2, Problem 29

We shall use partial integration:
int u dv=uv-int v du
Therefore, we have
int e^(-3x)sin5xdx=|[u=e^(-3x),dv=sin5xdx],[du=-3e^(-3x)dx,v=-1/5cos5x]|=
-1/5e^(-3x)cos5x-3/5int e^(-3x)cos5xdx=|[u=e^(-3x),dv=cos5xdx],[du=-3e^(-3x)dx,v=1/5sin5x]|=
-1/5e^(-3x)cos5x-3/25e^(-3x)sin5x-9/25inte^(-3x)sin5xdx
We can see that we have the same integral as the one we've started with. In other words we have the following equation
int e^(-3x)sin5xdx=-1/5e^(-3x)cos5x-3/25e^(-3x)sin5x
-9/25inte^(-3x)sin5xdx
Let us add 9/25int e^(-3x)sin5xdx to the whole equation.
34/25int e^(-3x)sin5xdx=-1/5e^(-3x)cos5x-3/25e^(-3x)sin5x
Now we only need to multiply the whole equation by 25/34 to obtain the solution to our starting problem.
int e^(-3x)sin5xdx=-5/34e^(-3x)cos5x-3/34e^(-3x)sin5x+c, c in RR

What did every Jew have to wear?

Elie Wiesel's Night is a semiautobiographical story, and, as such, the treatment to which the Jews are subjected in his book are a representation of the way the Nazis treated Jews in Romania, where Wiesel grew up before he was transported to the concentration camps along with his family.
Jews in Romania, as well as in Germany, Poland, and other countries subject to Hitler's demands, were expected to wear a yellow Star of David somewhere on their clothes. This star served to mark them out as different and to ensure that everyone who interacted with them knew the person wearing the star was Jewish. Refusal to wear the star was made against the law. Wearing the star also meant that Jews would be unable to break other rules, such as entering shops where the signage indicated that Jews were not welcome.