Differentiation, local behavior e^iπ = -1.

In this chapter we will finally see why $e^{i\pi }$ is $-1.$ 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 thetrigonometric and exponential functions. At this point, we only know their definitions as power series functions.The fact that ${sin}^2+{cos}^2=1$ or that $e^{x+y}=e^x{e}^y$ are not at all obvious. In fact, we haven't even yet defined what is meant by $e^x$ for an arbitrary number $x.$

The main theorems of this chapter include:

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