Summarize the following:
The chain rule is a rule for derivatives that you use when the original function combines a function within another function. The chain rule says that, for two functions f(x){\displaystyle f(x)} and g(x){\displaystyle g(x)}, the derivative of the combination of the two can be found as follows: If y=f(g(x)){\displaystyle y=f(g(x))}, then y′=f′(g)∗g′(x){\displaystyle y^{\prime }=f^{\prime }(g)*g^{\prime }(x)}. Using the chain rule requires that you first define the two functions that make up your combined function. For square root functions, the outer function f(g){\displaystyle f(g)} will be the square root function, and the inner function g(x){\displaystyle g(x)} will be whatever appears under the radical sign. For example, suppose you wish to find the derivative of 3x+2{\displaystyle {\sqrt {3x+2}}}. Define the two parts as follows:  f(g)=g=g12{\displaystyle f(g)={\sqrt {g}}=g^{\frac {1}{2}}} g(x)=(3x+2){\displaystyle g(x)=(3x+2)} To apply the chain rule to the square root of a function, you will first need to find the derivative of the general square root function:   f(g)=g=g12{\displaystyle f(g)={\sqrt {g}}=g^{\frac {1}{2}}}  f′(g)=12g−12{\displaystyle f^{\prime }(g)={\frac {1}{2}}g^{-{\frac {1}{2}}}} f′(g)=12g{\displaystyle f^{\prime }(g)={\frac {1}{2{\sqrt {g}}}}}   Then find the derivative of the second function:  g(x)=(3x+2){\displaystyle g(x)=(3x+2)} g′(x)=3{\displaystyle g^{\prime }(x)=3} Recall the chain rule, y′=f′(g)∗g′(x){\displaystyle y^{\prime }=f^{\prime }(g)*g^{\prime }(x)}, and then combine the derivatives as follows:  y′=12g∗3{\displaystyle y^{\prime }={\frac {1}{2{\sqrt {g}}}}*3} y′=12(3x+2∗3{\displaystyle y^{\prime }={\frac {1}{2{\sqrt {(3x+2}}}}*3} y′=32(3x+2{\displaystyle y^{\prime }={\frac {3}{2{\sqrt {(3x+2}}}}}

summary: Review the chain rule for functions. Define the functions for the chain rule. Find the derivatives of the two functions. Combine the functions in the chain rule.


Summarize the following:
Muslim countries are very concerned with respecting defined boundaries between the sexes.  If you are new to Pakistan and its culture, it is best to err on the side of caution when addressing the opposite sex.  Keep in mind that there are strict rules about men addressing women and about women addressing men.  Most Muslim women will not respond to greetings from men outside their families, and many men consider greetings from women, especially non-Muslim women, as very improper and impolite. The complicated Persian and Arabic origin dialects make Urdu a difficult language for non-native speakers.  The accent can vary between regions, but the most appropriate greeting when speaking with a Muslim is the Salam greeting.  Use the phrase "As-Salam-u-Alaikum", which means "Peace be unto you".  This phrase is pronounced “us-saa-laam-muu-alie-kum.” As with other languages, the pronouns in the Salam greeting will change depending on who you are greeting.  For example, the Salam greeting would be different if you are meeting a male business associate versus meeting the girlfriend of a colleague.  To modify the Salam greeting, you must change the “you” in the phrase, which is represented by the “-kum” part of As-Salam-u-Alaikum:  As-Salāmu `alayk(a): use when greeting one male As-Salāmu `alayk(i): use when greeting one female As-Salāmu `alayk(umā): use when greeting two people of any gender As-Salāmu `alayk(unna): use when greeting multiple females only' As-Salāmu `alayk(umu): use when greeting a group of three or more people, where at least one is a male or if you are meeting a member of state such as a prime minister, president, king, etc. Hierarchy is very important in Pakistan so greetings must be extended in a particular order.  This is especially true if you are meeting people for business.  Show respect by arriving on time and greeting the oldest person or the person with the highest position in the organization first.  Then greet people in descending order of age or position.  If you do not know everyone in the group, ask a mutual acquaintance to introduce you.  Do not introduce yourself because this is considered very rude.  Some other tips:  It is customary in Pakistan to need less personal space than many western cultures require, so don’t be surprised or back away if people stand very close to you while in a meeting. Only exchange business cards with either the right hand or with both hands.  Never use the left hand, as this is considered extremely rude. Make sure your business card lists your title and any advanced degrees in order to show your status.  If you are given a business card, be sure to show respect by studying the card and admiring their position and degrees before you place it into your cardholder. Because the conventions of propriety are much stricter in Muslim countries, you should base any physical greetings, like shaking hands or embracing, on the cues from the Muslim you are greeting.  If you are close with the person or they are of the middle class, shaking hands and embracing is much more common, even across sex lines.  Men typically shake hands with one another, and hugs are also common between Muslim and non-Muslim men if they have developed a relationship. Women will rarely hug or shake hands with men; however, some women in the middle and upper classes have adopted wearing gloves in order to bypass the strict law that says women can only make physical contact with male members of their family. Despite the strict rules governing the sexes, Pakistani culture is an incredibly social and vocal culture.  Once you have initiated the conversation with the Salam greeting, prepare for a longer conversation about the person's health, their family, and their business.  Show interest in the conversation, and don’t try to cut them off, as this is considered rude.
summary: Know the rules about addressing different sexes. Practice your pronunciation. Modify the greeting depending on your audience. Greet people in the appropriate order. Avoid physical contact unless it is initiated. Do not rush the conversation.