The business in AP Calc this week is all about the chain rule and anti derivatives. The chain rule has made it so that it seems like we can pretty much take the derivative of most functions we come across.
The chain rule is all about using breaking down a function into one fucntion inside of another to be able to break it down more easily. The basis of the chain rule is that f'(g(x))(g'(x)). The chain rule is useful when our other methods of finding derivatives fail. Like any other mathematical concept, it takes a few days to fully understand, but now it is yet another tool on the tool belt.
U-substitution was the other focus of this week. It is like the anti-derivative version of the chain rule. Substituting a part of the derivative function with u and using that to help see how to take the anti derivative. After we have a function that is u, we take the derivative of that and plug that back in to find the derivative in terms of u, instead of in terms of x. At the end of the problem we turn the function back into terms of x by plugging in the actual function at u.
The chain rule is all about using breaking down a function into one fucntion inside of another to be able to break it down more easily. The basis of the chain rule is that f'(g(x))(g'(x)). The chain rule is useful when our other methods of finding derivatives fail. Like any other mathematical concept, it takes a few days to fully understand, but now it is yet another tool on the tool belt.
U-substitution was the other focus of this week. It is like the anti-derivative version of the chain rule. Substituting a part of the derivative function with u and using that to help see how to take the anti derivative. After we have a function that is u, we take the derivative of that and plug that back in to find the derivative in terms of u, instead of in terms of x. At the end of the problem we turn the function back into terms of x by plugging in the actual function at u.