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 - 三角関数の微分

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)、三角関数の微分、問44、45、46.を取り組んでみる。

44. 
    $\begin{array}{l}\left(\cot x\right)\text{'}\\ =\left(\frac{1}{\tan x}\right)\text{'}\\ =\left(\frac{\cos x}{\sin x}\right)\text{'}\\ =\frac{-{\sin }^{2}x-{\cos }^{2}x}{{\sin }^{2}x}\\ =\frac{-1}{{\sin }^{2}x}\end{array}$
45. 1. 
       $3\cos 3x$
    2. 
       $-2\cos 2x$
    3. 
       $\frac{4}{{\cos }^{2}x}$
    4. 
       $2\sin x\cos x$
    5. 
       $-5{\cos }^{4}x\sin x$
    6. 
       $2\tan x·\frac{1}{{\cos }^{2}x}$
    7. 
       $4x\cos \left(2x^2-3\right)$
    8. 
       $\frac{4x^3-3x^2}{{\cos }^2\left(x^4-x^3\right)}$
    9. 
       ${\cos }^{2}x-{\sin }^{2}x$
    10. 
        $-\frac{3}{{\sin }^2\left(3x-4\right)}$
    11. 
        $6{\sin }^{2}2x\cos 2x$
    12. 
        $-\frac{1}{2}{\left(1+\cos x\right)}^{-\frac{1}{2}}\sin x$
    13. 
        $\frac{-\sin x\left(1-\sin x\right)+{\cos }^{2}x}{{\left(1-\sin x\right)}^2}$
    14. 
        $\frac{-\frac{1}{{\cos }^{2}x}\left(1+\tan x\right)-\left(1-\tan x\right)\frac{1}{{\cos }^{2}x}}{{\left(1+\tan x\right)}^2}$
46. 1. 
       $\begin{array}{l}y\text{'}=\cos x\\ \cos \frac{\pi }{4}=\frac{1}{\sqrt{2}}\end{array}$
    2. 
       $\begin{array}{l}y\text{'}=-\sin x\\ -\sin \frac{\pi }{6}=-\frac{1}{2}\end{array}$
    3. 
       $\begin{array}{l}y\text{'}={\cos }^{2}x-{\sin }^{2}x=1\\ {\cos }^2\frac{3\pi }{2}-{\sin }^2\frac{3\pi }{2}=-\frac{1}{2}-\frac{1}{2}=-1\end{array}$
    4. 
       $\begin{array}{l}y\text{'}=\frac{1}{{\cos }^{2}x}\\ \frac{1}{{\cos }^2\left(-\frac{\pi }{4}\right)}=2\end{array}$
    5. 
       $\begin{array}{l}y\text{'}=\frac{-\cos x}{{\sin }^{2}x}\\ \frac{-\cos \left(-\frac{\pi }{3}\right)}{{\sin }^2\left(-\frac{\pi }{3}\right)}=\frac{-\frac{1}{2}}{\frac{3}{4}}=-\frac{4}{2·3}=-\frac{2}{3}\end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, sin, cos, tan, pi, Derivative

print('46.')
x = symbols('x')

funcs = [(sin(x), pi / 4),
         (cos(x), pi / 6),
         (sin(x) * cos(x), 3 * pi / 2),
         (tan(x), - pi / 4),
         (1 / sin(x), -pi / 3)]

for i, (f, x0) in enumerate(funcs, 1):
    print('({})'.format(i))
    pprint(f)
    d = Derivative(f, x)
    f1 = d.doit()
    pprint(d)
    pprint(f1)
    pprint(f1.subs({x: x0}))

入出力結果(Terminal, IPython)

$ ./sample44.py
46.
(1)
sin(x)
d         
──(sin(x))
dx        
cos(x)
√2
──
2 
(2)
cos(x)
d         
──(cos(x))
dx        
-sin(x)
-1/2
(3)
sin(x)⋅cos(x)
d                
──(sin(x)⋅cos(x))
dx               
     2         2   
- sin (x) + cos (x)
-1
(4)
tan(x)
d         
──(tan(x))
dx        
   2       
tan (x) + 1
2
(5)
  1   
──────
sin(x)
d ⎛  1   ⎞
──⎜──────⎟
dx⎝sin(x)⎠
-cos(x) 
────────
   2    
sin (x) 
-2/3
$

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">
<br>
<label for="x0">x0 = </label>
<input id="x0" type="number" step="0.1" value="1">

<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="sample44.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'),
    input_x0 = document.querySelector('#x0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
             input_x0],
    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) => 1 / Math.tan(x),
    f1 = (x) => -1 / Math.sin(x) ** 2;

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),
        x0 = parseFloat(input_x0.value);
    
    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }

    let points = [],
        g = (x) => f1(x0) * (x - x0) + f(x0);
    
    for (let x = x1; x <= x2; x += dx) {
        let y = f(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    let lines = [[x1, g(x1), x2, g(x2)]];

    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, i) => i <= 1 ? 'black' : 'blue');
    
    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', 'red');
    
    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 =
x0 =