1.Differentiate (求微分） y = [ln(1+e^x)]^3

pick one's brains 意思是get information or ideas from someone（从某人那里获得消息、灵感等）.
有鼓励你去跟某人交流,交谈的意思在里面.
eg.I spent the afternoon with Donna,picking her brain for ideas to use in our celebration.

就是对(1/X)求导吧.等于负X的负二次方.

dy/dx=12x^2+2
dy/dx表示y对x的导数.

y = 1000000×[1.6^(¼x)]
= 1000000×e^{ln[1.6^(¼x)]}
= 1000000×e^[(¼x)ln1.6]
dy/dx = 1000000×{e^[(¼x)ln1.6]}×¼ln1.6
= 250000×{e^[(¼x)ln1.6]}×ln1.6
= 250000×[1.6^(¼x)]×ln1.6
≈117500.9073×[1.6^(¼x)]

第一题是求极限么?
lim x>无穷 [ln(8+x)-ln(5+x)]
=lim x→∞[ln(8+x)/(5+x)]
=ln[limx→∞ln(8+x)/(5+x)]
=ln1=0
∫20xe^(5x)dx
=4∫xde^5x
=4(xe^5x-∫e^5xdx)
=4(xe^5x-1/5*e^5x)+c
=(4x-1/5)e^5x+c
∫[(x-1)/(x^2+2x)]dx
=∫[-1/2*(1/x)+3/2*1/(x+2）]dx
=-1/2ln|x|+3/2ln|x+2|+c

纪录片与虚构片的差距到底有多大?

您好,
against 是一个副词,并不只有反对的意思,还有别的意思
作为附录我借鉴了别人的答案.请看第（6）条
这里是对照的意思
“他们不会对照着自己的目标来衡量自己的进度”
prep.（介词）
(1)
In a direction or course opposite to:
逆：与某个方向或过程相反：
row against the current.
逆水行舟
(2)
So as to come into forcible contact with:
对撞：为与…达成猛烈接触：
waves dashing against the shore.
拍岸的波涛
(3)
In contact with so as to rest or press on:
倚在；紧靠着：与…接触为了休息或按压：
leaned against the tree.
斜靠着树
(4)
In hostile opposition or resistance to:
反抗；抵抗：处于敌对状况或与…相抗衡：
struggle against fate.
同命运抗争
(5)
Contrary to; opposed to:
与…相反；对：
against my better judgment.
与我的判断相反
(6)
In contrast or comparison with the setting or background of:
对照；对比：以…的布景或背景相对比：
dark colors against a fair skin.
深色衬托出白皙的肌肤
(7)
In preparation for; in anticipation of:
预期；以备：为…作准备；期待着…：
food stored against winter.
为过冬贮备的食物
(8)
As a defense or safeguard from:
预防；防御：以…为防御或保护对象：
protection against the cold.
预防感冒
(9)
To the account or debt of:
兑换；提取：作为帐上的应收款项或作为…的借项：
drew a check against my bank balance.
开支票支取自己的银行结余
(10)
Directly opposite to; facing.
正对着；面向

答：
dy/dx=12x^2+2
dy/dx表示y对x的导数.
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首先啊,这个单词的引申意思是微分
做法是：
( (1- 2sinx)的导数 乘以 cosx + cosx的导数 乘以 (1- 2sinx) ) ÷ cos^2 x
得到的结果只要将（cos^2 x+sin^2 x）变成1,就得到这个答案sinx-2/cos^2 x.

已经逐渐地寻找区别提供独有类型的仿配方药的它的批发产品

Well, we can see their differences from two prospectives, namely, the scale and the concentration of power. Generally speaking, the one which has one or more branches oversea can be called multinational company. While, it could never name itself as a global company unless it could find its way into almost evevry country in the world; On the other hand, we can see that multinational company share powers and put a lot of efforts into localization, as to the global companies, they do the opposite things.

dy={1/[2sqrt(x+sqrt(x+(sqrt(x)))]} d[x+sqrt(x+(sqrt(x)))]
={1/[2sqrt(x+sqrt(x+(sqrt(x)))]} {dx+1/[2sqrt(x)+(sqrt(x))]d[x+(sqrt(x))]}
={1/[2sqrt(x+sqrt(x+(sqrt(x)))]} {dx+1/[2sqrt(x)+(sqrt(x))][dx+1/(2sqrt(x))dx]}
={1/[2sqrt(x+sqrt(x+(sqrt(x)))]} {1+1/[2sqrt(x)+(sqrt(x))][1+1/(2sqrt(x))]}dx
即
Differentiate y={1/[2sqrt(x+sqrt(x+(sqrt(x)))]} {1+1/[2sqrt(x)+(sqrt(x))][1+1/(2sqrt(x))]}

前者,是
后者,不一定.
单变量情况下,求微分跟求导数一样
多变量情况下,不同.

term,名词,术语,用语
自然科学这个术语也用于区别那些使用科学的方法来研究自然的科研领域.

differentiation,微分（名词）
differentiate,求导,或者求微分（动词）
differential（微分的）形容词
还有一个：differentiable可微的