Chapter 7 : Transcendental Functions

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博客围绕函数相关知识展开，介绍了反函数及其导数，包括一一对应函数、反函数及可微函数反函数的导数；还涉及自然对数、对数求导法，以及指数函数中e^x的导数、积分和e的极限表示，此外提及指数增长与衰减等内容。

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7.1 Inverse Functions and Their Derivatives

One-to-One Functions

Inverse Functions

Since each output of a one-to-one function comes from just one input, the effect of the function can be inverted to send an output back to the input from which it came.

Derivatives of Inverses of Differentiable Functions

7.2 Natural Logarithms

Logarithmic Differentiation

The derivatives of positive functions given by formulas that involve products, quotients, and powers can often be found more quickly if we take the natural logarithm of both sides before differentiating. This enables us to use the laws of logarithms to simplify the formulas before differentiating. The process, called logarithmic differentiation, is illustrated in the next example.

7.3 The Exponential Functions

The Derivative and Integral of e^x

The Number e Expressed as a Limit

We have defined the number e as the number for which or the value exp (1). We see that e is an important constant for the logarithmic and exponential functions, but what is its numerical value? The next theorem shows one way to calculate e as a limit.