数学 - Python - 関数の変化をとらえる - 関数の極限と微分法 – いろいろな関数の導関数 - 高次導関数(三角関数(正弦、余弦)、対数関数、合成関数、指数関数、帰納法)

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)、17.5(いろいろな関数の導関数)、高次導関数、問58、59、60、61.を取り組んでみる。

58. 1. 
       $\begin{array}{l}y\text{'}=5{\left(x^2+1\right)}^{4}2x\\ y\text{'}\text{'}=20{\left(x^2+1\right)}^{3}4x^2+5{\left(x^2+1\right)}^{4}2\end{array}$
    2. 
       $y\text{'}\text{'}\text{'}=6$
    3. 
       $f^{\left(10\right)}\left(x\right)=0$
    4. 
       $f^{\left(4\right)}\left(x\right)=2^4{e}^{2x}=16e^{2x}$
    5. 
       $f^{\left(7\right)}\left(x\right)=f^{\left(3\right)}\left(x\right)=-\cos x$
    6. 
       $f^{\left(10\right)}\left(x\right)=f^{\left(2\right)}\left(x\right)=-\sin x$
    7. 
       $f^{\left(2\right)}\left(x\right)=-\cos x$
    8. 
       $f^{\left(11\right)}\left(x\right)=f^{\left(3\right)}\left(x\right)=\sin x$
    9. 
       $f^{\left(2\right)}\left(x\right)=\frac{-1}{x^2}$
    10. 
        $\begin{array}{l}f\text{'}\left(x\right)=\frac{-1}{x^2}\\ f\text{'}\text{'}\left(x\right)=\frac{2x}{x^4}=\frac{2}{x^3}\\ f^{\left(3\right)}\left(x\right)=\frac{-2·3x^2}{x^6}=-\frac{6}{x^4}\\ f^{\left(4\right)}\left(x\right)=\frac{6·4x^3}{x^8}=\frac{24}{x^5}\end{array}$
59. 1. 
       $\begin{array}{l}\left(f\text{'}g+fg\text{'}\right)\text{'}\\ =f\text{'}\text{'}g+f\text{'}g\text{'}+f\text{'}g\text{'}+fg\text{'}\text{'}\\ =f\text{'}\text{'}g+2f\text{'}g\text{'}+fg\text{'}\text{'}\end{array}$
    2. 
       $\begin{array}{l}f\text{'}\text{'}\text{'}g+f\text{'}\text{'}g\text{'}+2f\text{'}\text{'}g\text{'}+2f\text{'}g\text{'}\text{'}+f\text{'}g\text{'}\text{'}+fg\text{'}\text{'}\text{'}\\ =f\text{'}\text{'}\text{'}g+3f\text{'}\text{'}g\text{'}+3f\text{'}g\text{'}\text{'}+fg\text{'}\text{'}\text{'}\end{array}$
    3. 
       ${\displaystyle \sum _{i=0}^n{f}^{\left(n-i\right)}{g}^{\left(i\right)}}$
60. 
    $\begin{array}{l}{\left(-1\right)}^0\left(x-0\right)e^{-x}=x{e}^{-x}\\ \\ {\left(-1\right)}^n\left(e^{-x}+\left(x-n\right)e^{-x}\left(-1\right)\right)\\ ={\left(-1\right)}^{n+1}\left(-1+x-n\right)e^{-x}\\ ={\left(-1\right)}^{n+1}\left(x-\left(n+1\right)\right)e^{-x}\end{array}$
61. 
    $\begin{array}{l}\frac{d}{dx}\left(x^0\log x\right)=\frac{d}{dx}\log x=\frac{1}{x}=\frac{0!}{x}\\ \\ \frac{d^{k+2}}{d{x}^{k+2}}\left(x^{k+1}\log x\right)\\ =\frac{d^{k+1}}{d{x}^{k+1}}\left(\frac{d}{dx}\left(x^{k+1}\log x\right)\right)\\ =\frac{d^{k+1}}{d{x}^{k+1}}\left(\left(k+1\right)x^k\log x+x^k\right)\\ =\frac{d^{k+1}}{d{x}^{k+1}}\left(\left(k+1\right)x^k\log x\right)+\frac{d^{k+1}}{d{x}^{k+1}}{x}^k\\ =\left(k+1\right)\frac{d^{k+1}}{d{x}^{k+1}}\left(x^k\log x\right)\\ =\left(k+1\right)\frac{k!}{x}\\ =\frac{\left(k+1\right)!}{x}\end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, sin, cos, log, Derivative, Function

x = symbols('x')
print('58.')

funcs = [(sin(x), 10),
         (cos(x), 11),
         (log(x), 2),
         (1 / x, 4)]

for func, n in funcs:
    d = Derivative(func, x, n)
    fn = d.doit()
    pprint(d)
    pprint(fn)
    print()

print('59.')
f = Function('f')
g = Function('g')

fg = f(g(x))
for n in [2, 3]:
    d = Derivative(fg, x, n)
    fn = d.doit()
    pprint(d)
    pprint(d.doit())
    print()

入出力結果(Terminal, IPython)

$ ./sample58.py
58.
 10         
d           
────(sin(x))
  10        
dx          
-sin(x)

 11         
d           
────(cos(x))
  11        
dx          
sin(x)

  2        
 d         
───(log(x))
  2        
dx         
-1 
───
  2
 x 

  4   
 d ⎛1⎞
───⎜─⎟
  4⎝x⎠
dx    
24
──
 5
x 

59.
  2         
 d          
───(f(g(x)))
  2         
dx          
          2    2                2                           
⎛d       ⎞    d                d        ⎛ d        ⎞│       
⎜──(g(x))⎟ ⋅──────(f(g(x))) + ───(g(x))⋅⎜───(f(ξ₁))⎟│       
⎝dx      ⎠       2              2       ⎝dξ₁       ⎠│ξ₁=g(x)
            dg(x)             dx                            

  3         
 d          
───(f(g(x)))
  3         
dx          
          3    3                            2              2           3           
⎛d       ⎞    d                 d          d              d           d        ⎛ d 
⎜──(g(x))⎟ ⋅──────(f(g(x))) + 3⋅──(g(x))⋅──────(f(g(x)))⋅───(g(x)) + ───(g(x))⋅⎜───
⎝dx      ⎠       3              dx            2            2           3       ⎝dξ₁
            dg(x)                        dg(x)           dx          dx            

                
       ⎞│       
(f(ξ₁))⎟│       
       ⎠│ξ₁=g(x)
                

$

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="sample58.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 f = (x) => x ** 3 + 3 * x - 6,
    f1 = (x) => 3 * x ** 2 + 3,
    f2 = (x) => 6 * x,
    f3 = (x) => 6;

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 = [];
    
    for (let x = x1; x <= x2; x += dx) {
        let y = f(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y, 'red']);
        }
    }
    let t1 = points.length;
    for (let x = x1; x <= x2; x += dx) {
        let y = f1(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y, 'green']);
        }
    }    
    let lines = [];

    for (let x = x1; x <= x2; x += 100 * dx) {
        lines.push([x1, f2(x1), x2, f2(x2), 'blue']);
    }
    for (let x = x1; x <= x2; x += 100 * dx) {
        lines.push([x1, f3(x1), x2, f3(x2), 'brown']);
    }

    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('line')
        .data([[x1, 0, x2, 0], [0, y1, 0, y2]].concat(lines))
        .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', (d) => d[4] || 'black');
    
    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', (d) => d[2] || 'green');
    
    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 =