反雙曲函數的微分

You’ll find this post in your _posts directory. Go ahead and edit it and re-build the site to see your changes. You can rebuild the site in many different ways, but the most common way is to run jekyll serve, which launches a web server and auto-regenerates your site when a file is updated.

To add new posts, simply add a file in the _posts directory that follows the convention YYYY-MM-DD-name-of-post.ext and includes the necessary front matter. Take a look at the source for this post to get an idea about how it works.

Jekyll also offers powerful support for code snippets:

def print_hi(name)
  puts "Hi, #{name}"
end
print_hi('Tom')
#=> prints 'Hi, Tom' to STDOUT.

Check out the Jekyll docs for more info on how to get the most out of Jekyll. File all bugs/feature requests at Jekyll’s GitHub repo. If you have questions, you can ask them on Jekyll Talk.

以下要介紹常見的反雙曲函數的微分方法 (導函數) , 我會仔細撰寫他的詳細證明過程，而再開始證明前 , 你還需要知道雙曲函數的微分以及一些不等式，我再下面都會一一說明

你必須先知道的

雙曲函數的微分

$\frac{d}{dx}(sinh(x))=cosh(x)$
$\frac{d}{dx}(cosh(x))=sinh(x)$
$\frac{d}{dx}(tanh(x))=sech^2(x)$
$\frac{d}{dx}(sech(x))=-sech(x)tanh(x)$
$\frac{d}{dx}(csch(x))=-csch(x)coth(x)$
$\frac{d}{dx}(coth(x))=-csch^2(x)$

雙曲函數恆等式

$cosh^2(x)-sinh^2(x)=1$

反雙曲函數詳細微分過程

1. $sinh^{-1}(x)$ 的微分

令 $y=sinh^{-1}(x)\Rightarrow sinh(y)=x$

$\frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}=\frac{1}{cosh(y)}$

$\because cosh(x)=\sqrt{1+sinh^2(y)}$

$\therefore \frac{1}{cosh(y)}=\frac{1}{\sqrt{1+sinh^2(y)}}$

最後再將 $x$ 代入 , 可得 $\frac{1}{\sqrt{1+x^2}}$

得證 $\Rightarrow \frac{d}{dx}(sinh^{-1}(x))=\frac{1}{\sqrt{1+x^2}}$

2. $cosh^{-1}(x)$ 的微分

令 $y=cosh^{-1}(x)\Rightarrow cosh(y)=x$

$\frac{dy}{dx}=\frac{1}{\frac{dx}{dy}}=\frac{1}{sinh(y)}$

$\because sinh(y)=\sqrt{cosh^2(y)-1}$

$\therefore \frac{1}{sinh(y)}=\frac{1}{\sqrt{cosh^2(y)-1}}$

最後再將 $x$ 代入 , 可得 $\frac{1}{\sqrt{x^2-1}}$

得證 $\Rightarrow \frac{d}{dx}(cosh^{-1}(x))=\frac{1}{\sqrt{x^2-1}}$

未完待續…

This is a test