## 2017年10月6日金曜日

### 数学 - Python - JavaScript - 合成関数、逆関数の微分法(導関数、ニュートン商の極限)( @hyuki @morinobukuzunu1 )結城浩さんのツイート、葛貫森信（くずにゃん）さんリツイートより

ということで、早速定義に立ち返って合成関数の微分、逆関数の微分、導関数を求めてみる。

$\begin{array}{l}\frac{d}{dx}f\left(g\left(x\right)\right)\\ =\underset{h\to 0}{\mathrm{lim}}\frac{f\left(g\left(x+h\right)\right)-f\left(g\left(x\right)\right)}{h}\\ =\underset{h\to 0}{\mathrm{lim}}\frac{f\left(g\left(x+h\right)-g\left(x\right)+g\left(x\right)\right)-f\left(g\left(x\right)\right)}{h}\\ =\underset{h\to 0}{\mathrm{lim}}\frac{f\left(g\left(x\right)+\left(g\left(x+h\right)-g\left(x\right)\right)\right)-f\left(g\left(x\right)\right)}{h}\\ =\underset{h\to 0}{\mathrm{lim}}\frac{f\left(g\left(x\right)+\left(g\left(x+h\right)-g\left(x\right)\right)\right)-f\left(g\left(x\right)\right)}{h}·\frac{g\left(x+h\right)-g\left(x\right)}{g\left(x+h\right)-g\left(x\right)}\\ =\underset{h\to 0}{\mathrm{lim}}\frac{f\left(g\left(x\right)+\left(g\left(x+h\right)-g\left(x\right)\right)\right)-f\left(g\left(x\right)\right)}{g\left(x+h\right)-g\left(x\right)}·\frac{g\left(x+h\right)-g\left(x\right)}{h}\\ =\underset{h\to 0}{\mathrm{lim}}\frac{f\left(g\left(x\right)+\left(g\left(x+h\right)-g\left(x\right)\right)\right)-f\left(g\left(x\right)\right)}{g\left(x+h\right)-g\left(x\right)}\underset{h\to 0}{\mathrm{lim}}\frac{g\left(x+h\right)-g\left(x\right)}{h}\\ =\underset{g\left(x+h\right)-g\left(x\right)\to 0}{\mathrm{lim}}\frac{f\left(g\left(x\right)+\left(g\left(x+h\right)-g\left(x\right)\right)\right)-f\left(g\left(x\right)\right)}{g\left(x+h\right)-g\left(x\right)}\underset{h\to 0}{\mathrm{lim}}\frac{g\left(x+h\right)-g\left(x\right)}{h}\\ =f\text{'}\left(g\left(x\right)\right)g\text{'}\left(x\right)\\ \\ \frac{d}{dx}f\left(g\left(x\right)\right)=f\text{'}\left(g\left(x\right)\right)g\text{'}\left(x\right)\end{array}$

$\begin{array}{l}y=f\left(x\right)\\ x={f}^{-1}\left(y\right)\\ \\ \frac{dx}{dy}=\underset{h\to 0}{\mathrm{lim}}\frac{{f}^{-1}\left(y+h\right)-{f}^{-1}\left(y\right)}{h}\\ =\underset{h\to 0}{\mathrm{lim}}\frac{{f}^{-1}\left(y+h\right)-{f}^{-1}\left(y\right)}{h+y-y}\\ =\underset{h\to 0}{\mathrm{lim}}\frac{{f}^{-1}\left(y+h\right)-{f}^{-1}\left(y\right)}{y+h-f\left(x\right)}\\ =\underset{h\to 0}{\mathrm{lim}}\frac{{f}^{-1}\left(y+h\right)-{f}^{-1}\left(y\right)}{f\left({f}^{-1}\left(y+h\right)\right)-f\left(x\right)}\\ =\underset{h\to 0}{\mathrm{lim}}\frac{{f}^{-1}\left(y+h\right)-{f}^{-1}\left(y\right)}{f\left({f}^{-1}\left(y+h\right)+{f}^{-1}\left(y\right)-{f}^{-1}\left(y\right)\right)-f\left(x\right)}\\ =\underset{h\to 0}{\mathrm{lim}}\frac{{f}^{-1}\left(y+h\right)-{f}^{-1}\left(y\right)}{f\left({f}^{-1}\left(y+h\right)+x-{f}^{-1}\left(y\right)\right)-f\left(x\right)}\\ =\underset{h\to 0}{\mathrm{lim}}\frac{{f}^{-1}\left(y+h\right)-{f}^{-1}\left(y\right)}{f\left(x+{f}^{-1}\left(y+h\right)-{f}^{-1}\left(y\right)\right)-f\left(x\right)}\\ =\underset{h\to 0}{\mathrm{lim}}\frac{{f}^{-1}\left(y+h\right)-{f}^{-1}\left(y\right)}{f\left(x+\left({f}^{-1}\left(y+h\right)-{f}^{-1}\left(y\right)\right)\right)-f\left(x\right)}\\ =\underset{h\to 0}{\mathrm{lim}}\frac{1}{\frac{f\left(x+\left({f}^{-1}\left(y+h\right)-{f}^{-1}\left(y\right)\right)\right)-f\left(x\right)}{{f}^{-1}\left(y+h\right)-{f}^{-1}\left(y\right)}}\\ =\frac{1}{\underset{h\to 0}{\mathrm{lim}}\frac{f\left(x+\left({f}^{-1}\left(y+h\right)-{f}^{-1}\left(y\right)\right)\right)-f\left(x\right)}{{f}^{-1}\left(y+h\right)-{f}^{-1}\left(y\right)}}\\ =\frac{1}{\underset{{f}^{-1}\left(y+h\right)-{f}^{-1}\left(y\right)\to 0}{\mathrm{lim}}\frac{f\left(x+\left({f}^{-1}\left(y+h\right)-{f}^{-1}\left(y\right)\right)\right)-f\left(x\right)}{{f}^{-1}\left(y+h\right)-{f}^{-1}\left(y\right)}}\\ =\frac{1}{f\text{'}\left(x\right)}\\ =\frac{1}{\frac{dy}{dx}}\end{array}$

SymPy(Python)の極限(Limit関数)と微分(Derivative関数)で確認。

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, Function, Derivative, Limit

print('合成関数の微分法')
f = Function('f')
g = Function('g')
x, h = symbols('x h')

for dir in ['+', '-']:
l = Limit((f(g(x + h)) - f(g(x))) / h, h, 0, dir=dir)
for t in [l, l.doit()]:
pprint(t)
print()
print()

gf = g(f(x))
Dx = Derivative(gf, x, 1)
for t in [Dx, Dx.doit()]:
pprint(t)
print()


$./sample.py 合成関数の微分法 ⎛-f(g(x)) + f(g(h + x))⎞ lim ⎜──────────────────────⎟ h─→0⁺⎝ h ⎠ ⎛ d ⎞│ ⎛ d ⎞│ ⎜───(f(ξ₁))⎟│ ⋅⎜───(g(ξ₁))⎟│ ⎝dξ₁ ⎠│ξ₁=g(x) ⎝dξ₁ ⎠│ξ₁=x ⎛-f(g(x)) + f(g(h + x))⎞ lim ⎜──────────────────────⎟ h─→0⁻⎝ h ⎠ ⎛ d ⎞│ ⎛ d ⎞│ ⎜───(f(ξ₁))⎟│ ⋅⎜───(g(ξ₁))⎟│ ⎝dξ₁ ⎠│ξ₁=g(x) ⎝dξ₁ ⎠│ξ₁=x d ──(g(f(x))) dx d ⎛ d ⎞│ ──(f(x))⋅⎜───(g(ξ₁))⎟│ dx ⎝dξ₁ ⎠│ξ₁=f(x)$


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="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'),
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.exp(x),
f1 = (x) => Math.exp(x),
g = (x) => Math.log(x),
g1 = (x) => 1 / f1(g(x)),
h1 = (x0) => (x) => f1(x0) * (x - x0) + f(x0),
h2 = (x0) => (x) => g1(x0) * (x - x0) + g(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),
dx0 = parseFloat(input_dx0.value);

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

let points = [],
lines = [],
fns = [[f, 'green'],
[g, 'blue']],
fns1 = [[(x) => x, 'red']],
fns2 = [[h1, 'orange'],
[h2, 'brown']];

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

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

fns1
.forEach((o) => {
let [f, color] = o;

lines.push([x1, f(x1), x2, f(x2), 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();