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 - 累乗の微分(対数微分法)

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, Google Chrome...)用JavaScript Library: MathJax
参考書籍

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)、累乗の微分、問54.を取り組んでみる。

1. $\begin{array}{l} \log y = \log x \log x \\ \log y = {(\log x)}^{2} \\ \frac{y'}{y} = \frac{2}{x} \log x \\ y' = \frac{2 y}{x} \log x = \frac{2 x^{\log x}}{x} \log x = 2 x^{\log x - 1} \log x \end{array}$
2. $\begin{array}{l} \log y = x^{x} \log x \\ \frac{y'}{y} = \frac{d}{d x} (x^{x}) \log x + x^{x} \frac{1}{x} \\ y' = x^{(x^{x})} (\frac{d}{d x} (x^{x}) \log x + x^{x - 1}) \\ z = x^{x} \\ \log z = x \log x \\ \frac{z'}{z} = \log x + 1 \\ z' = x^{x} (\log x + 1) \\ y' = x^{(x^{x})} (x^{x} (\log x + 1) \log x + x^{x - 1}) \end{array}$

コード(Emacs)

Python 3

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

from sympy import pprint, symbols, log, Derivative, factor, expand

x = symbols('x')
print('54.')

funcs = [x ** log(x), x ** (x ** x)]

for i, func in enumerate(funcs, 1):
    print('({})'.format(i))
    pprint(func)
    d = Derivative(func, x)
    f1 = d.doit()
    pprint(f1)
    pprint(factor(f1))
    pprint(expand(f1))

入出力結果(Terminal, IPython)

$ ./sample54.py
54.
(1)
 log(x)
x      
   log(x)       
2⋅x      ⋅log(x)
────────────────
       x        
   log(x)       
2⋅x      ⋅log(x)
────────────────
       x        
   log(x)       
2⋅x      ⋅log(x)
────────────────
       x        
(2)
 ⎛ x⎞
 ⎝x ⎠
x    
 ⎛ x⎞ ⎛                          x⎞
 ⎝x ⎠ ⎜ x                       x ⎟
x    ⋅⎜x ⋅(log(x) + 1)⋅log(x) + ──⎟
      ⎝                         x ⎠
    ⎛ x⎞                           
 x  ⎝x ⎠ ⎛     2                  ⎞
x ⋅x    ⋅⎝x⋅log (x) + x⋅log(x) + 1⎠
───────────────────────────────────
                 x                 
                                         ⎛ x⎞
    ⎛ x⎞               ⎛ x⎞           x  ⎝x ⎠
 x  ⎝x ⎠    2       x  ⎝x ⎠          x ⋅x    
x ⋅x    ⋅log (x) + x ⋅x    ⋅log(x) + ────────
                                        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="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="sample54.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) => x ** Math.log(x),
    f1 = (x) => 2 * x ** Math.log(x - 1) * Math.log(x),
    g = (x) => x ** (x ** x),
    g1 = (x) =>  x ** (x ** x) * (x ** x * (Math.log(x) + 1) * Math.log(x) + x ** (x - 1)),
    h = (f, f1, x0) => (x) => f1(x0) * (x - x0) + f(x0);

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 = [],
        fh = h(f, f1, x0),
        gh = h(g, g1, x0);
    
    for (let x = x1; x <= x2; x += dx) {
        let y = f(x);

        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    let t = points.length;
    
    for (let x = x1; x <= x2; x += dx) {
        let y = g(x);
        if (x === -1) {
            console.log('hit');
        }
        if (Math.abs(y) < Infinity) {
            points.push([x, y]);
        }
    }
    let lines = [[x1, fh(x1), x2, fh(x2), 'blue'],
                 [x1, gh(x1), x2, gh(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, i) => i < t ? 'red' : '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 =
x0 =

Kamimura's blog

2017年6月8日木曜日

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 - 累乗の微分(対数微分法)

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