怎么求函数的分数阶导数？兼谈数学手册计算器

目前，很少有数学软件可以计算函数的分数阶导数。但是，小巧的mathhandbook（数学手册计算器）却可以。想知道怎么办到的吗，那就请往下看。Few math-software can workout the fractional order derivative of functi . But the compact mathhandbook (MathHandBook Calculator) can do. If You want to know how can it do, Please look down.

1尝试着用Mathematica来计算正弦函数的1/2阶导数，无果。I try to use Mathematica to calculate the 1/2 orderderivative of sine function, but no result.D[Sin[x],{x,1/2}]

2但是，数学手册却可以办到。But the MathHandBook can do.先输入sin(x)，然后点击“半阶导数”按钮，就可以看到结果。Clicksin(x),thenclickthe “semi derivative” buttontogetresult .

3所以，这里暂时忽略Mathematica。计算sin(x)的1/3阶导数，就没有固定的按钮可以使用了，需要自己写代码，然后按回车。So, Mathematica is ignored here for the moment.Solve the 1/3 order derivativeof sin(x), there is no fixed button can be used, You need to write your own code, and then press enter.The code reads as below:d(sin(x),x,1/3)

4如果要考虑较一般的情形，可以尝试用下面这段代码：If You want to find a general case, You can try the following code:d(sin(x),x,1/n)

5不过上面代码的运行结果有点问题，实际上是下图所示。However, the run-result of the ve code has a minor mistake and is actually shown in the following fig .

6更一般的情形。More general situation:d(sin(x),x,m/n)

7能不能求出无理数阶导数呢？比如计算正弦函数的sqrt(2)阶导数。Can it get the irrational number order derivative?For example, the sqrt(2) order derivative of the sine function .d(sin(x),x,sqrt(2))运行结果采用了浮点数表示的：The results are represented by floating-point numbers:sin(0.7071067811865476*pi+x)而这个常数0.7071067811865476恰好就是sqrt(2)/2的前16位小数形式。And this c tant 0.7071067811865476 happens to be the first 16 Decimal forms of sqrt (2) /2.

8正弦函数的π阶导数和1/π阶导数。The π order derivative and the 1/π orderderivative of sine function.d(sin(x),x,pi)d(sin(x),x,1/pi)

大家猜一下，正弦函数的sqrt(-1)阶导数会是什么呢？

Let's guess, what is the sqrt(-1) order derivative of a sine function?