数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 逆関数 - 逆正接関数(逆正弦関数、逆余弦関数、導関数、合成関数の微分法、積の微分法、分数関数の微分法)

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, Microsoft Edge, Google Chrome...)用JavaScript Library: MathJax
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7章(逆関数)、4(逆正接関数)、練習問題5、6、7、8、9、10、11、12、13、14、15、16.を取り組んでみる。

$\begin{array}{l} \frac{d}{d x} (\arctan 3 x) \\ = \frac{1}{1 + 9 x^{2}} \cdot 3 \\ = \frac{3}{1 + 9 x^{2}} \end{array}$
$\begin{array}{l} \frac{d}{d x} (\arctan \sqrt{x}) \\ = \frac{1}{1 + x^{2}} \cdot \frac{1}{2} x^{- \frac{1}{2}} \\ = \frac{1}{2 (1 + x^{2}) \sqrt{x}} \end{array}$
$\begin{array}{l} \frac{d}{d x} (\arcsin x + \arccos) \\ = \frac{1}{\sqrt{1 - x^{2}}} - \frac{1}{\sqrt{1 - x^{2}}} \\ = 0 \end{array}$
$\begin{array}{l} \frac{d}{d x} (x \arcsin x) \\ = \arcsin x + x \frac{1}{\sqrt{1 - x^{2}}} \\ = \arcsin x + \frac{x}{\sqrt{1 - x^{2}}} \end{array}$
$\begin{array}{l} \frac{d}{d x} (\arcsin (\sin (2 x))) \\ = \frac{1}{\sqrt{1 - \sin^{2} (2 x)}} \cdot \cos (2 x) \cdot 2 \\ = \frac{2 \cos (2 x)}{\sqrt{1 - \sin^{2} (2 x)}} \end{array}$
$\begin{array}{l} \frac{d}{d x} (x^{2} \arctan 2 x) \\ = 2 x \arctan 2 x + x^{2} \frac{1}{1 + 4 x^{2}} \cdot 2 \\ = 2 x \arctan 2 x + \frac{2 x^{2}}{1 + 4 x^{2}} \end{array}$
$\begin{array}{l} \frac{d}{d x} (\frac{\sin x}{\arcsin x}) \\ = \frac{\cos x \arcsin x - \sin x \frac{1}{\sqrt{1 - x^{2}}}}{\arcsin^{2} x} \\ = \frac{\sqrt{1 - x^{2}} \cos x \arcsin x - \sin x}{\sqrt{1 - x^{2}} \arcsin^{2} x} \end{array}$
$\begin{array}{l} \frac{d}{d x} (\arcsin (\cos x - x^{2})) \\ = \frac{1}{\sqrt{1 - {(\cos x - x^{2})}^{2}}} \cdot (- \sin x - 2 x) \\ = - \frac{\sin x + 2 x}{\sqrt{1 - {(\cos x - x^{2})}^{2}}} \end{array}$
$\begin{array}{l} \frac{d}{d x} (\arctan \frac{1}{x}) \\ = \frac{1}{1 + \frac{1}{x^{2}}} \cdot \frac{- 1}{x^{2}} \\ = - \frac{1}{1 + x^{2}} \end{array}$
$\begin{array}{l} \frac{d}{d x} (\arctan \frac{1}{2 x}) \\ = \frac{1}{1 + \frac{1}{4 x^{2}}} \cdot \frac{- 2}{4 x^{2}} \\ = - \frac{2}{1 + 4 x^{2}} \end{array}$
$\begin{array}{l} \frac{d}{d x} {(1 + \arcsin 3 x)}^{3} \\ = 3 {(1 + \arcsin 3 x)}^{2} \cdot \frac{1}{\sqrt{1 - 9 x^{2}}} \cdot 3 \\ = \frac{9 {(1 + \arcsin 3 x)}^{2}}{\sqrt{1 - 9 x^{2}}} \end{array}$
$\begin{array}{l} \frac{d}{d x} {(\arcsin 2 x + \arctan x^{2})}^{\frac{3}{2}} \\ = \frac{3}{2} {(\arcsin 2 x + \arctan x^{2})}^{\frac{1}{2}} (\frac{1}{\sqrt{1 - 4 x^{2}}} \cdot 2 + \frac{1}{1 + x^{4}} \cdot 2 x) \\ = 3 \sqrt{\arcsin 2 x + \arctan x^{2}} (\frac{1}{\sqrt{1 - 4 x^{2}}} + \frac{x}{1 + x^{4}}) \end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, sin, cos, asin, acos, atan, Derivative, sqrt

x = symbols('x')
fs = [atan(3 * x),
      atan(sqrt(x)),
      asin(x) + acos(x),
      x * asin(x),
      asin(sin(2 * x)),
      x ** 2 * atan(2 * x),
      sin(x) / asin(x),
      asin(cos(x) - x ** 2),
      atan(1 / x),
      atan(1 / 2 * x),
      (1 + asin(3 * x)) ** 3,
      sqrt((asin(2 * x) + atan(x ** 2)) ** 3)]

for i, f in enumerate(fs, 5):
    print(f'{i}.')
    D = Derivative(f, x, 1)
    for t in [D, D.doit()]:
        pprint(t.factor())
        print()
    print()

入出力結果(Terminal, Jupyter(IPython))

$ ./sample5.py
5.
d            
──(atan(3⋅x))
dx           

   3    
────────
   2    
9⋅x  + 1


6.
d           
──(atan(√x))
dx          

     1      
────────────
2⋅√x⋅(x + 1)


7.
d                    
──(acos(x) + asin(x))
dx                   

0


8.
d            
──(x⋅asin(x))
dx           

      x                
───────────── + asin(x)
   __________          
  ╱    2               
╲╱  - x  + 1           


9.
d                 
──(asin(sin(2⋅x)))
dx                

     2⋅cos(2⋅x)     
────────────────────
   _________________
  ╱      2          
╲╱  - sin (2⋅x) + 1 


10.
d ⎛ 2          ⎞
──⎝x ⋅atan(2⋅x)⎠
dx              

     2                  
  2⋅x                   
──────── + 2⋅x⋅atan(2⋅x)
   2                    
4⋅x  + 1                


11.
d ⎛ sin(x)⎞
──⎜───────⎟
dx⎝asin(x)⎠

 cos(x)           sin(x)        
─────── - ──────────────────────
asin(x)      __________         
            ╱    2          2   
          ╲╱  - x  + 1 ⋅asin (x)


12.
d ⎛     ⎛ 2         ⎞⎞
──⎝-asin⎝x  - cos(x)⎠⎠
dx                    

     -(2⋅x + sin(x))      
──────────────────────────
    ______________________
   ╱                2     
  ╱    ⎛ 2         ⎞      
╲╱   - ⎝x  - cos(x)⎠  + 1 


13.
d ⎛    ⎛1⎞⎞
──⎜atan⎜─⎟⎟
dx⎝    ⎝x⎠⎠

    -1     
───────────
 2 ⎛    1 ⎞
x ⋅⎜1 + ──⎟
   ⎜     2⎟
   ⎝    x ⎠


14.
d              
──(atan(0.5⋅x))
dx             

    0.5    
───────────
      2    
0.25⋅x  + 1


15.
d ⎛               3⎞
──⎝(asin(3⋅x) + 1) ⎠
dx                  

                 2
9⋅(asin(3⋅x) + 1) 
──────────────────
    ____________  
   ╱      2       
 ╲╱  - 9⋅x  + 1   


16.
  ⎛    _________________________⎞
  ⎜   ╱                       3 ⎟
d ⎜  ╱  ⎛                ⎛ 2⎞⎞  ⎟
──⎝╲╱   ⎝asin(2⋅x) + atan⎝x ⎠⎠  ⎠
dx                               

                               _________________________
                              ╱                       3 
⎛ 6⋅x            6       ⎞   ╱  ⎛                ⎛ 2⎞⎞  
⎜────── + ───────────────⎟⋅╲╱   ⎝asin(2⋅x) + atan⎝x ⎠⎠  
⎜ 4          ____________⎟                              
⎜x  + 1     ╱      2     ⎟                              
⎝         ╲╱  - 4⋅x  + 1 ⎠                              
────────────────────────────────────────────────────────
                  ⎛                ⎛ 2⎞⎞                
                2⋅⎝asin(2⋅x) + atan⎝x ⎠⎠                


iMac:dir4 kamimura$ ./sample5.py
5.
d            
──(atan(3⋅x))
dx           

   3    
────────
   2    
9⋅x  + 1


6.
d           
──(atan(√x))
dx          

     1      
────────────
2⋅√x⋅(x + 1)


7.
d                    
──(acos(x) + asin(x))
dx                   

0


8.
d            
──(x⋅asin(x))
dx           

       __________        
      ╱    2             
x + ╲╱  - x  + 1 ⋅asin(x)
─────────────────────────
     __________________  
   ╲╱ -(x - 1)⋅(x + 1)   


9.
d                 
──(asin(sin(2⋅x)))
dx                

            2⋅cos(2⋅x)            
──────────────────────────────────
  ________________________________
╲╱ -(sin(2⋅x) - 1)⋅(sin(2⋅x) + 1) 


10.
d ⎛ 2          ⎞
──⎝x ⋅atan(2⋅x)⎠
dx              

    ⎛   2                          ⎞
2⋅x⋅⎝4⋅x ⋅atan(2⋅x) + x + atan(2⋅x)⎠
────────────────────────────────────
                 2                  
              4⋅x  + 1              


11.
d ⎛ sin(x)⎞
──⎜───────⎟
dx⎝asin(x)⎠

   __________                        
  ╱    2                             
╲╱  - x  + 1 ⋅cos(x)⋅asin(x) - sin(x)
─────────────────────────────────────
      __________________     2       
    ╲╱ -(x - 1)⋅(x + 1) ⋅asin (x)    


12.
d ⎛     ⎛ 2         ⎞⎞
──⎝-asin⎝x  - cos(x)⎠⎠
dx                    

             -(2⋅x + sin(x))             
─────────────────────────────────────────
   ______________________________________
  ╱  ⎛ 2             ⎞ ⎛ 2             ⎞ 
╲╱  -⎝x  - cos(x) - 1⎠⋅⎝x  - cos(x) + 1⎠ 


13.
d ⎛    ⎛1⎞⎞
──⎜atan⎜─⎟⎟
dx⎝    ⎝x⎠⎠

 -1   
──────
 2    
x  + 1


14.
d              
──(atan(0.5⋅x))
dx             

     0.5     
─────────────
      2      
0.25⋅x  + 1.0


15.
d ⎛    3              2                       ⎞
──⎝asin (3⋅x) + 3⋅asin (3⋅x) + 3⋅asin(3⋅x) + 1⎠
dx                                             

                    2   
   9⋅(asin(3⋅x) + 1)    
────────────────────────
  ______________________
╲╱ -(3⋅x - 1)⋅(3⋅x + 1) 


16.
  ⎛   ________________________________________________________________________
d ⎜  ╱     3              2          ⎛ 2⎞                   2⎛ 2⎞       3⎛ 2⎞ 
──⎝╲╱  asin (2⋅x) + 3⋅asin (2⋅x)⋅atan⎝x ⎠ + 3⋅asin(2⋅x)⋅atan ⎝x ⎠ + atan ⎝x ⎠ 
dx                                                                            

⎞
⎟
⎠
 

                                   _________________________
  ⎛          ____________    ⎞    ╱                       3 
  ⎜ 4       ╱      2         ⎟   ╱  ⎛                ⎛ 2⎞⎞  
3⋅⎝x  + x⋅╲╱  - 4⋅x  + 1  + 1⎠⋅╲╱   ⎝asin(2⋅x) + atan⎝x ⎠⎠  
────────────────────────────────────────────────────────────
    ______________________ ⎛ 4    ⎞ ⎛                ⎛ 2⎞⎞  
  ╲╱ -(2⋅x - 1)⋅(2⋅x + 1) ⋅⎝x  + 1⎠⋅⎝asin(2⋅x) + atan⎝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="-5">
<label for="x2">x2 = </label>
<input id="x2" type="number" value="5">
<br>
<label for="y1">y1 = </label>
<input id="y1" type="number" value="-5">
<label for="y2">y2 = </label>
<input id="y2" type="number" value="5">
<br>
<label for="dx0">dx0 = </label>
<input id="dx0" type="number" min="0" step="0.0001" value="0.1">

<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="sample5.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_dx0 = document.querySelector('#dx0'),
    inputs = [input_r, input_dx, input_x1, input_x2, input_y1, input_y2,
              input_dx0],
    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) => Math.asin(x),
    g = (x) => Math.acos(x),
    h = (x) => Math.atan(x),
    f1 = (x) => 1 / Math.sqrt(1 - x ** 2),
    g1 = (x) => -1 / Math.sqrt(1 - x ** 2),
    h1 = (x) => 1 / (1 + 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),
        dx0 = parseFloat(input_dx0.value);

    if (r === 0 || dx === 0 || x1 > x2 || y1 > y2) {
        return;
    }    

    let points = [],
        lines = [],        
        fns = [[f, 'red'],
               [g, 'green'],
               [h, 'blue'],
               [f1, 'orange'],
               [g1, 'brown'],
               [h1, 'skyblue']],
        fns1 = [],
        fns2 = [];

    fns
        .forEach((o) => {
            let [f, color] = o;
            for (let x = x1; x <= x2; x += dx) {
                let y = f(x);

                points.push([x, y, color]);
            }
        });

    fns2
        .forEach((o) => {
            let [f, color] = o;

            for (let x = x1; x <= x2; x += dx0) {
                let g = f(x);
                lines.push([x1, g(x1), x2, g(x2), color]);
            }
        });
    
    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);

    [fns, fns1, fns2].forEach((fs) => p(fs.join('\n')));
};

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

r = dx =
x1 = x2 =
y1 = y2 =
dx0 =

Kamimura's blog

2017年10月2日月曜日

数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 逆関数 - 逆正接関数(逆正弦関数、逆余弦関数、導関数、合成関数の微分法、積の微分法、分数関数の微分法)

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