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# Class Contents

## Exponential Functions

We are now in a position to use the amazing relationship between exponentiation and power series to prove a number of important results very quickly. In the next class we shall use the exponential functions to define and analyze the trigonometric functions.`e`^{x}, exponentials and logarithms with this base play a key part in pure mathematics. They are so central to analysis, that we write
`e`^{x} is differentiable on the real line, and that it is equal to its derivative. We shall extend the result to differentiating general exponential functions:

The first thing we shall do is show that the logarithm base `e` is differentiable. This is a straightforward application of the
Inverse Function Theorem.

Remark:

Since the key to understanding all exponential maps is the special map
for the logarithm base `e` (i.e. we don't need a subscript `e`). This is also the reason that logarithms base `e` are called **natural logarithms**.