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# Differentiation, Local Behavior E^iπ = -1.

Module by: Lawrence Baggett. E-mail the author

Summary: This is an overview of important topics on differentiation, including links to theorems from the same author.

In this chapter we will finally see why eiπeiπ is -1.-1. Along the way, we will give careful proofs of all the standard theorems of Differential Calculus, and in the process we will discover all the familiar facts about the trigonometric and exponential functions. At this point, we only know their definitions as power series functions. The fact that sin2+cos2=1sin2+cos2=1 or that ex+y=exeyex+y=exey are not at all obvious. In fact, we haven't even yet defined what is meant by exex for an arbitrary number x.x.

The main theorems of this chapter include:

1. The Chain Rule ((Reference)),
2. The Mean Value Theorem ((Reference)),
3. The Inverse Function Theorem ((Reference)),
4. The Laws of Exponents ((Reference) and (Reference)), and
5. Taylor's Remainder Theorem ((Reference)).

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