数学 - Python - JavaScript - 関数の変化をとらえる - 関数の極限と微分法 – 導関数とその計算 - 積および商の微分(商の微分の定理とニュートン商)

数学読本〈4〉

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
• 参考書籍
  • Pythonからはじめる数学入門
  • JavaScriptリファレンス 第6版
  • インタラクティブ・データビジュアライゼーション

数学読本〈4〉数列の極限，順列/順列・組合せ/確率/関数の極限と微分法(松坂 和夫(著)、岩波書店)の第17章(関数の変化をとらえる - 関数の極限と微分法)、17.3(導関数とその計算)、積および商の微分、問30.を取り組んでみる。

30. 1. 
       $\frac{3}{{\left(2-x\right)}^2}$
    2. 
       $\frac{x^2+1-x·2x}{{\left(x^2+1\right)}^2}$
    3. 
       $\frac{2x\left(3x+4\right)-\left(x^2+2\right)3}{{\left(3x+4\right)}^2}$
    4. 
       $\frac{\left(2x-2\right)\left(x^2+x+2\right)-\left(x^2-2x+6\right)\left(2x+1\right)}{{\left(x^2+x+2\right)}^2}$
    5. 
       $\frac{2x\left(x^3-4\right)-\left(x^2+3\right)3x^2}{{\left(x^3-4\right)}^2}$
    6. 
       $\frac{3{\left(3x+2\right)}^{2}3-{\left(3x+2\right)}^{3}2\left(2x-1\right)2}{{\left(2x-1\right)}^4}$

コード(Emacs)

Python 3

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

from sympy import symbols, Derivative, pprint, factor

x = symbols('x')

print('30.')
exprs = [
    (3, 2 - x),
    (x, x ** 2 + 1),
    (x ** 2 + 2, 3 * x + 4),
    (x ** 2 - 2 * x + 6, x ** 2 + x + 2),
    (x ** 2 + 3, x ** 3 - 4),
    ((3 * x + 2) ** 3, (2 * x - 1) ** 2)
]

for i, (num, den) in enumerate(exprs, 1):
    print('({0})'.format(i))
    expr = num / den
    pprint(expr)
    d = Derivative(expr, x)
    pprint(d)
    d = d.doit()
    pprint(d)
    pprint(factor(d))

入出力結果(Terminal, IPython)

$ ./sample30.py
30.
(1)
  3   
──────
-x + 2
d ⎛  3   ⎞
──⎜──────⎟
dx⎝-x + 2⎠
    3    
─────────
        2
(-x + 2) 
   3    
────────
       2
(x - 2) 
(2)
  x   
──────
 2    
x  + 1
d ⎛  x   ⎞
──⎜──────⎟
dx⎜ 2    ⎟
  ⎝x  + 1⎠
        2           
     2⋅x        1   
- ───────── + ──────
          2    2    
  ⎛ 2    ⎞    x  + 1
  ⎝x  + 1⎠          
-(x - 1)⋅(x + 1) 
─────────────────
            2    
    ⎛ 2    ⎞     
    ⎝x  + 1⎠     
(3)
  2    
 x  + 2
───────
3⋅x + 4
  ⎛  2    ⎞
d ⎜ x  + 2⎟
──⎜───────⎟
dx⎝3⋅x + 4⎠
            ⎛ 2    ⎞
  2⋅x     3⋅⎝x  + 2⎠
─────── - ──────────
3⋅x + 4            2
          (3⋅x + 4) 
   2          
3⋅x  + 8⋅x - 6
──────────────
           2  
  (3⋅x + 4)   
(4)
 2          
x  - 2⋅x + 6
────────────
  2         
 x  + x + 2 
  ⎛ 2          ⎞
d ⎜x  - 2⋅x + 6⎟
──⎜────────────⎟
dx⎜  2         ⎟
  ⎝ x  + x + 2 ⎠
           ⎛ 2          ⎞             
(-2⋅x - 1)⋅⎝x  - 2⋅x + 6⎠    2⋅x - 2  
───────────────────────── + ──────────
                  2          2        
      ⎛ 2        ⎞          x  + x + 2
      ⎝x  + x + 2⎠                    
   2           
3⋅x  - 8⋅x - 10
───────────────
             2 
 ⎛ 2        ⎞  
 ⎝x  + x + 2⎠  
(5)
 2    
x  + 3
──────
 3    
x  - 4
  ⎛ 2    ⎞
d ⎜x  + 3⎟
──⎜──────⎟
dx⎜ 3    ⎟
  ⎝x  - 4⎠
     2 ⎛ 2    ⎞         
  3⋅x ⋅⎝x  + 3⎠    2⋅x  
- ───────────── + ──────
            2      3    
    ⎛ 3    ⎞      x  - 4
    ⎝x  - 4⎠            
   ⎛ 3          ⎞ 
-x⋅⎝x  + 9⋅x + 8⎠ 
──────────────────
            2     
    ⎛ 3    ⎞      
    ⎝x  - 4⎠      
(6)
         3
(3⋅x + 2) 
──────────
         2
(2⋅x - 1) 
  ⎛         3⎞
d ⎜(3⋅x + 2) ⎟
──⎜──────────⎟
dx⎜         2⎟
  ⎝(2⋅x - 1) ⎠
           2              3
9⋅(3⋅x + 2)    4⋅(3⋅x + 2) 
──────────── - ────────────
          2              3 
 (2⋅x - 1)      (2⋅x - 1)  
         2           
(3⋅x + 2) ⋅(6⋅x - 17)
─────────────────────
               3     
      (2⋅x - 1)
$      

HTML5

<div id="graph0"></div>
<pre id="output0"></pre>
<label for="r0">r = </label>
<input id="r0" type="number" min="0" value="0.5">
<label for="dx">dx = </label>
<input id="dx" type="number" min="0" step="0.0001" value="0.001">
<br>
<label for="x1">x1 = </label>
<input id="x1" type="number" value="-10">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="10">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-10">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="10">

<button id="draw0">draw</button>
<button id="clear0">clear</button>

<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/d3/4.2.6/d3.min.js" integrity="sha256-5idA201uSwHAROtCops7codXJ0vja+6wbBrZdQ6ETQc=" crossorigin="anonymous"></script>

<script src="sample30.js"></script>    

JavaScript

let div0 = document.querySelector('#graph0'),
    pre0 = document.querySelector('#output0'),
    width = 600,
    height = 600,
    padding = 50,
    btn0 = document.querySelector('#draw0'),
    btn1 = document.querySelector('#clear0'),
    input_r = document.querySelector('#r0'),
    input_dx = document.querySelector('#dx'),
    input_x1 = document.querySelector('#x1'),
    input_x2 = document.querySelector('#x2'),
    input_y1 = document.querySelector('#y1'),
    input_y2 = document.querySelector('#y2'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2],

    p = (x) => pre0.textContent += x + '\n',
    range = (start, end, step=1) => {
        let res = [];
        for (let i = start; i < end; i += step) {
            res.push(i);
        }
        return res;
    };

let draw = () => {
    pre0.textContent = '';

    let r = parseFloat(input_r.value),
        dx = parseFloat(input_dx.value),
        x1 = parseFloat(input_x1.value),
        x2 = parseFloat(input_x2.value),
        y1 = parseFloat(input_y1.value),
        y2 = parseFloat(input_y2.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        f = (x) => (3 * x + 2) ** 3 / (2 * x - 1) ** 2
    
    for (let x = x1; x <= x2; x += dx) {
        let y = f(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    
    let xscale = d3.scaleLinear()
        .domain([x1, x2])
        .range([padding, width - padding]);
    let yscale = d3.scaleLinear()
        .domain([y1, y2])
        .range([height - padding, padding]);

    let xaxis = d3.axisBottom().scale(xscale);
    let yaxis = d3.axisLeft().scale(yscale);
    div0.innerHTML = '';
    let svg = d3.select('#graph0')
        .append('svg')
        .attr('width', width)
        .attr('height', height);

    svg.selectAll('circle')
        .data(points)
        .enter()
        .append('circle')
        .attr('cx', (d) => xscale(d[0]))
        .attr('cy', (d) => yscale(d[1]))
        .attr('r', r)
        .attr('fill', 'green');

    svg.selectAll('line')
        .data([[x1, 0, x2, 0], [0, y1, 0, y2]])
        .enter()
        .append('line')
        .attr('x1', (d) => xscale(d[0]))
        .attr('y1', (d) => yscale(d[1]))
        .attr('x2', (d) => xscale(d[2]))
        .attr('y2', (d) => yscale(d[3]))
        .attr('stroke', 'black');
    
    svg.append('g')
        .attr('transform', `translate(0, ${height - padding})`)
        .call(xaxis);

    svg.append('g')
        .attr('transform', `translate(${padding}, 0)`)
        .call(yaxis);
};

inputs.forEach((input) => input.onchange = draw);
btn0.onclick = draw;
btn1.onclick = () => pre0.textContent = '';
draw();

r = dx =
x1 = x2 =
y1 = y2 =