Problem: Article: 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^{
rac {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^{
rac {1}{2}}} f(g)=12g12{\displaystyle f^{\prime }(g)={
rac {1}{2}}g^{-{
rac {1}{2}}}} f(g)=12g{\displaystyle f^{\prime }(g)={
rac {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=12g3{\displaystyle y^{\prime }={
rac {1}{2{\sqrt {g}}}}*3} y=12(3x+23{\displaystyle y^{\prime }={
rac {1}{2{\sqrt {(3x+2}}}}*3} y=32(3x+2{\displaystyle y^{\prime }={
rac {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.

Problem: Article: About half an hour prior to takeoff, the gate attendant will announce the boarding time. Boarding is done in sections, either by group/sections (designated with a letter) or by rows/seats. Check your boarding pass to see if you belong to a group, and if not, wait for your row or a section of rows to be called. First class always boards the plane first, followed by business class and people with disabilities or infants. Although it is not always the case, it may be beneficial to try to snag a place in line early on so that there is still room in the overhead bins for your luggage. Otherwise, if storage space runs out your bags will have to be checked. After waiting in line for boarding, there will be a gate attendant next to the entrance waiting to check your boarding pass. If you are flying nationally, hand your boarding pass to the gate attendant who will scan your boarding pass in (by placing it down on the terminal), but you will not need your ID out at this point. If you are flying internationally, you will need to show your passport again at this point. Keep your boarding pass out after having it checked, as you may need to show it to an additional flight attendant on the plane. While most hallways are short, there are some gates who may have line hallways much longer and curvier than others. There is typically a buildup of passengers after the boarding pass check, so you may have to wait in line again before boarding the plane. Check your seat to make sure you are headed in the right direction, and keep a lookout for your row number. If you need help finding your seat, ask others around which seats are for which row and later which seat letter is for which seat (there is no "one size fits all" for explanations here). Most flight attendants are busy getting their supplies ready for your flight. If you still require more help finding your seat, step into an open seat near the row and once the flight attendant comes around to check on your seat belt, you can ask them then.) When you have located your seat, place your smaller bag under your chair, and look for available space in the overhead bins for your larger carry-on. This isn't always the easiest task, so turn to a flight attendant for assistance in searching or placing your bag if necessary. When you're finally able to get in your seat, place your smaller carry-on under the seat in front of you. You're all done! Now it's time for you to sit back and relax while you are carried away to your destination in luxury. During the flight, you will sometimes be offered a free drink and food (depending on the carrier and length of the flight). If needed, there are restrooms in the front and back of the plane. Any other questions can be directed towards flight attendants.
Summary:
Wait for the announcement to board. Get your boarding pass checked. Walk down the hallway that leads up to your plane. Enter the aircraft. Stow your carry-on items. Get settled in.