3.5 The Chain Rule

Hey everyone, I know that this may say that Ryan is posting the blog but it is in fact I, Alex, who am posting the blog under Ryan's account. Unfortunately I was rather inept when it came to making my blog account, so here I am. Before I get into the notes taken from today's class, I would like to take this opportunity to remind Izzy that his day as blog master is tomorrow. Anywho, onto the lesson!

Today we learned the chain rule, which, to put it simply, is taking all of the differentiability rules that we have learned so far and applying them to composite functions and power functions. There are a few really important formulas to remember here, one of which is the basic formula for the chain rule. Given that the function F(x)=f(g(x)), then,



In other words, the derivative of the composite function f(g(x)) equals the derivative of f(x) times the function g(x) times the derivative of g(x)

Example problem:

Differentiate the following:



We can take this concept of the chain rule a step further and apply it to the power rule. The power rule's general equation is as follows if n is any real number and u=g(x) is differentiable:



Another way to state this equation is:



Example Problem:

Differentiate the following:



And now we can apply all of this knowledge to find the Derivative of the Exponential!! This may seem somewhat daunting because Oh No! It uses a natural logarithm! (well that's at least how I think-you may think differently), but it is substantially less confusing than it may seem. Here is the general equation for the Derivative of the exponential:



In other words, to find the derivative of a given number "a" raised to any exponent "x", you multiply that number raised to the exponent by the natural log of the number

Example Problem:

Differentiate the following:



Well, that's pretty much it for all the notes and whatnot. Now enjoy my little artistic creation that I have here. I can't help enjoying it, because Grey's Anatomy is the best show on television.Bye everyone!! IZZY YOU GO TOMORROW!! Thought that I should just remind you again :-)