数学 - Python - SymPy - 解析学 - 微分と基本的な関数 - 微分係数、導関数(和、積、および商, Derivative, doit method)

解析入門 原書第3版
原著

学習環境

• Surface 3 (4G LTE)、Surface 3 タイプ カバー、Surface ペン(端末)
• Windows 10 Pro (OS)
• 数式入力ソフト(TeX, MathML): MathType
• MathML対応ブラウザ: Firefox、Safari
• MathML非対応ブラウザ(Internet Explorer, Google Chrome...)用JavaScript Library: MathJax
• Pythonからはじめる数学入門(参考書籍)

解析入門 原書第3版 (S.ラング(著)、松坂 和夫(翻訳)、片山 孝次(翻訳)、岩波書店)の第2部(微分と基本的な関数)、第3章(微分係数、導関数)、補充問題 21-40.を取り組んでみる。

21. 
    $\frac{-2\left(x+1\right)+2x}{{\left(x+1\right)}^2}$
22. 
    $\frac{x-5-\left(x+1\right)}{{\left(x-5\right)}^2}$
23. 
    $\frac{\frac{1}{2}{x}^{-\frac{1}{2}}\left(x+1\right)\left(x-1\right)-x^{\frac{1}{2}}\left(x-1+x+1\right)}{\left(x+1\right){\left(x-1\right)}^2}$
24. 
    $\frac{\left(x^{-\frac{1}{2}}+\frac{3}{4}{x}^{-\frac{1}{4}}\right)-\left(2x^{\frac{1}{2}}+x^{\frac{3}{4}}\right)\left(x^3+\left(x+1\right)3x^2\right)}{{\left(\left(x+1\right)x^3\right)}^2}$
25. 
    $\frac{5x^4\left(x^2+1\right)\left(x+7\right)-\left(x^5+1\right)\left(2x\left(x+7\right)+\left(x^2+1\right)\right)}{{\left(x^2+1\right)}^2{\left(x+7\right)}^2}$
26. 
    $\frac{\left(x+5+x+1\right)\left(x-4\right)-\left(x+1\right)\left(x+5\right)}{{\left(x-4\right)}^2}$
27. 
    $\frac{3x^2\left(1-x^2\right)-x^3\left(-2x\right)}{{\left(1-x^2\right)}^2}$
28. 
    $\frac{5x^4\left(x^{\frac{3}{2}}+x\right)-x^5\left(\frac{3}{2}{x}^{\frac{1}{2}}+1\right)}{{\left(x^{\frac{3}{2}}+x\right)}^2}$
29. 
    $\frac{\left(2x=1\right)\left(x^2+1\right)-\left(x^2-x\right)2x}{{\left(x^2+1\right)}^2}$
30. 
    $\frac{\left(2x+2\right)8x-\left(x^2+2x+7\right)8}{64x^2}$
31. 
    $\frac{2\left(x^2+x-4\right)-\left(2x+1\right)\left(2x+1\right)}{{\left(x^2+x-4\right)}^2}$
32. 
    $\frac{5x^4\left(x^2+3\right)-x^{5}2x}{{\left(x^2+3\right)}^2}$
33. 
    $\frac{\left(4-3x^2\right)\left(x^2+2\right)-\left(4x-x^3\right)2x}{{\left(x^2+2\right)}^2}$
34. 
    $\frac{3x^2\left(x^2-5x+7\right)-x^3\left(2x-5\right)}{{\left(x^2-5x+7\right)}^2}$
35. 
    $\frac{-5x-\left(1-5x\right)}{x^2}$
36. 
    $\frac{\left(6+\frac{3}{4}{x}^{-\frac{1}{4}}\right)\left(7x-2\right)-\left(1+6x+x^{\frac{3}{4}}\right)7}{{\left(7x-2\right)}^2}$
37. 
    $\frac{2x\left(x+1\right)\left(x-2\right)-x^2\left(x-2+x+1\right)}{{\left(x+1\right)}^2{\left(x-2\right)}^2}$
38. 
    $\frac{\left(\frac{1}{2}{x}^{-\frac{1}{2}}+\frac{1}{2}{x}^{-\frac{3}{2}}\right)x^{\frac{3}{4}}-\left(x^{\frac{1}{2}}-x^{-\frac{1}{2}}\right)\frac{3}{2}{x}^{\frac{1}{2}}}{x^{\frac{3}{2}}}$
39. 
    $\frac{\left(12x^3+\frac{5}{4}{x}^{\frac{1}{4}}\right)\left(4x^3-x^5+1\right)-\left(3x^4+x^{\frac{5}{4}}\right)\left(12x^2-5x^4\right)}{{\left(4x^3-x^5+1\right)}^2}$
40. 
    $\frac{\left(x-2\right)\left(x-3\right)-\left(x-1\right)\left(x-3+x-2\right)}{{\left(x-2\right)}^2{\left(x-3\right)}^2}$

コード(Emacs)

#!/usr/bin/env python3
# -*- coding: utf-8 -*-

from sympy import Symbol, pprint, Derivative

x = Symbol('x')
exprs = [
    x ** 2 / ((x + 1) * (x - 2)),
    (x ** 1 / 2 - x ** (-1 / 2)) / x ** (3 / 4),
    (3 * x ** 4 + x ** (5 / 4)) / (4 * x ** 3 - x ** 5 + 1),
    (x - 1) / ((x - 2) * (x - 3))
]
for i, expr in enumerate(exprs):
    print('{0}.'.format(i + 37))
    pprint(Derivative(expr).doit())

入出力結果(Terminal, IPython)

$ ./sample21.py
37.
          2                  2                         
         x                  x                 2⋅x      
- ──────────────── - ──────────────── + ───────────────
                 2          2           (x - 2)⋅(x + 1)
  (x - 2)⋅(x + 1)    (x - 2) ⋅(x + 1)                  
38.
        -1.75 ⎛   -0.5   x⎞    -0.75 ⎛     -1.5   1⎞
- 0.75⋅x     ⋅⎜- x     + ─⎟ + x     ⋅⎜0.5⋅x     + ─⎟
              ⎝          2⎠          ⎝            2⎠
39.
      0.25       3   ⎛   4    1.25⎞ ⎛   4       2⎞
1.25⋅x     + 12⋅x    ⎝3⋅x  + x    ⎠⋅⎝5⋅x  - 12⋅x ⎠
────────────────── + ─────────────────────────────
    5      3                                2     
 - x  + 4⋅x  + 1           ⎛   5      3    ⎞      
                           ⎝- x  + 4⋅x  + 1⎠      
40.
       1               x - 1              x - 1      
─────────────── - ──────────────── - ────────────────
(x - 3)⋅(x - 2)                  2          2        
                  (x - 3)⋅(x - 2)    (x - 3) ⋅(x - 2)
$