## 2017年10月30日月曜日

### 数学 - Python - JavaScript - 解析学 - 微分と基本的な関数 - 指数関数と対数関数 - 指数関数(三角関数(正弦、余弦、正接、逆正弦、逆余弦、逆正接)、累乗(べき乗))

1. $\frac{1}{1+{\left(\mathrm{log}x\right)}^{2}}·\frac{1}{x}$

2. $\begin{array}{l}\frac{1}{\mathrm{cos}\left(3x+5\right)}\left(-\mathrm{sin}\left(3x+5\right)\right)3\\ =-3\mathrm{tan}\left(3x+5\right)\end{array}$

3. ${e}^{\mathrm{sin}\left(2x\right)}\mathrm{cos}\left(2x\right)2=2{e}^{\mathrm{sin}\left(2x\right)}\mathrm{cos}\left(2x\right)$

4. ${e}^{\mathrm{arccos}x}\frac{-1}{\sqrt{1-{x}^{2}}}=-\frac{{e}^{\mathrm{arccos}x}}{\sqrt{1-{x}^{2}}}$

5. $\frac{1}{{e}^{x}}{e}^{x}=1$

6. $\frac{{e}^{x}-x{e}^{x}}{{e}^{2x}}=\frac{1-x}{{e}^{x}}$

7. ${e}^{{e}^{x}}·{e}^{x}={e}^{{e}^{x}+x}$

8. $\begin{array}{l}{e}^{-\mathrm{arcsin}x}\frac{-1}{\sqrt{1-{x}^{2}}}\\ =-\frac{{e}^{-\mathrm{arcsin}x}}{\sqrt{1-{x}^{2}}}\end{array}$

9. $\frac{{e}^{x}}{{\mathrm{cos}}^{2}\left({e}^{x}\right)}$

10. $\begin{array}{l}y={x}^{\sqrt{x}}\\ \mathrm{log}y=\mathrm{log}{x}^{\sqrt{x}}\\ \mathrm{log}y=\sqrt{x}\mathrm{log}x\\ \frac{d}{dx}\mathrm{log}y=\frac{1}{2}{x}^{-\frac{1}{2}}\mathrm{log}x+\sqrt{x}·\frac{1}{x}\\ \frac{d}{dy}\left(\mathrm{log}y\right)·\frac{dy}{dx}=\frac{1}{2}{x}^{-\frac{1}{2}}\mathrm{log}x+\sqrt{x}·\frac{1}{x}\\ \frac{1}{y}\frac{dy}{dx}=\frac{1}{2}{x}^{-\frac{1}{2}}\mathrm{log}x+{x}^{-\frac{1}{2}}\\ \frac{dy}{dx}=\left(\frac{1}{2}\mathrm{log}x+1\right){x}^{-\frac{1}{2}}y\\ \frac{dy}{dx}=\left(\frac{1}{2}\mathrm{log}x+1\right){x}^{-\frac{1}{2}}{x}^{\sqrt{x}}\\ \frac{dy}{dx}=\left(\frac{1}{2}\mathrm{log}x+1\right){x}^{\sqrt{x}-\frac{1}{2}}\end{array}$

11. $\begin{array}{l}y={x}^{{x}^{\frac{1}{3}}}\\ \mathrm{log}y=\mathrm{log}{x}^{{x}^{\frac{1}{3}}}\\ \mathrm{log}y={x}^{\frac{1}{3}}\mathrm{log}x\\ \frac{d}{dx}\mathrm{log}y=\frac{1}{3}{x}^{-\frac{2}{3}}\mathrm{log}x+{x}^{\frac{1}{3}}\frac{1}{x}\\ \frac{d}{dy}\mathrm{log}y·\frac{dy}{dx}=\frac{1}{3}{x}^{-\frac{2}{3}}\mathrm{log}x+{x}^{-\frac{2}{3}}\\ \frac{1}{y}\frac{dy}{dx}=\left(\frac{1}{3}\mathrm{log}x+1\right){x}^{-\frac{2}{3}}\\ \frac{dy}{dx}=\left(\frac{1}{3}\mathrm{log}x+1\right){x}^{-\frac{2}{3}}y\\ \frac{dy}{dx}=\left(\frac{1}{3}\mathrm{log}x+1\right){x}^{-\frac{2}{3}}{x}^{{x}^{\frac{1}{3}}}\\ \frac{dy}{dx}=\left(\frac{1}{3}\mathrm{log}x+1\right){x}^{{x}^{\frac{1}{3}}-\frac{2}{3}}\end{array}$

12. $\frac{{e}^{x}+1}{\sqrt{1-{\left({e}^{x}+x\right)}^{2}}}$

13. ${e}^{\mathrm{tan}x}·\frac{1}{{\mathrm{cos}}^{2}x}=\frac{{e}^{\mathrm{tan}x}}{{\mathrm{cos}}^{2}x}$

14. $\frac{{e}^{x}}{1+{e}^{2x}}$

コード(Emacs)

Python 3

#!/usr/bin/env python3
from sympy import pprint, symbols, exp, log, sin, cos, tan, asin, acos, atan, Rational, sqrt, Derivative, plot

print('5.')
x = symbols('x')
fs = [atan(log(x)),
log(cos(3 * x + 5)),
exp(sin(2 * x)),
exp(acos(x)),
log(exp(x)),
x / exp(x),
exp(exp(x)),
exp(-asin(x)),
tan(exp(x)),
x ** sqrt(x),
x ** (x ** Rational(1 / 3)),
asin(exp(x) + x),
exp(tan(x)),
atan(exp(x))]

for i, f in enumerate(fs):
c = chr(ord("a") + i)
print(f'({c})')
try:
D = Derivative(f, x, 1)
f1 = D.doit()
for t in [D, f1]:
pprint(t)
print()
print()
p = plot(f, f1, show=False, legend=True)
for j, color in enumerate(['red', 'green']):
p[j].line_color = color
p.save(f'sample5_{c}.png')
except Exception as err:
print(type(err), err)


$./sample5.py 5. (a) d ──(atan(log(x))) dx 1 ─────────────── ⎛ 2 ⎞ x⋅⎝log (x) + 1⎠ (b) d ──(log(cos(3⋅x + 5))) dx -3⋅sin(3⋅x + 5) ──────────────── cos(3⋅x + 5) (c) d ⎛ sin(2⋅x)⎞ ──⎝ℯ ⎠ dx sin(2⋅x) 2⋅ℯ ⋅cos(2⋅x) (d) d ⎛ acos(x)⎞ ──⎝ℯ ⎠ dx acos(x) -ℯ ───────────── __________ ╱ 2 ╲╱ - x + 1 (e) d ⎛ ⎛ x⎞⎞ ──⎝log⎝ℯ ⎠⎠ dx 1 (f) d ⎛ -x⎞ ──⎝x⋅ℯ ⎠ dx -x -x - x⋅ℯ + ℯ (g) ⎛ ⎛ x⎞⎞ d ⎜ ⎝ℯ ⎠⎟ ──⎝ℯ ⎠ dx ⎛ x⎞ x ⎝ℯ ⎠ ℯ ⋅ℯ /opt/local/Library/Frameworks/Python.framework/Versions/3.6/lib/python3.6/site-packages/sympy/plotting/experimental_lambdify.py:232: UserWarning: The evaluation of the expression is problematic. We are trying a failback method that may still work. Please report this as a bug. warnings.warn('The evaluation of the expression is' /opt/local/Library/Frameworks/Python.framework/Versions/3.6/lib/python3.6/site-packages/sympy/plotting/plot.py:1109: RuntimeWarning: invalid value encountered in double_scalars cos_theta = dot_product / (vector_a_norm * vector_b_norm) /opt/local/Library/Frameworks/Python.framework/Versions/3.6/lib/python3.6/site-packages/sympy/plotting/plot.py:1105: RuntimeWarning: invalid value encountered in subtract vector_b = (z - y).astype(np.float) /opt/local/Library/Frameworks/Python.framework/Versions/3.6/lib/python3.6/site-packages/sympy/plotting/plot.py:1104: RuntimeWarning: invalid value encountered in subtract vector_a = (x - y).astype(np.float) /opt/local/Library/Frameworks/Python.framework/Versions/3.6/lib/python3.6/site-packages/sympy/plotting/plot.py:1007: RuntimeWarning: overflow encountered in double_scalars pos_bottom = ('data', 0) if yl*yh <= 0 else 'center' (h) d ⎛ -asin(x)⎞ ──⎝ℯ ⎠ dx -asin(x) -ℯ ───────────── __________ ╱ 2 ╲╱ - x + 1 (i) d ⎛ ⎛ x⎞⎞ ──⎝tan⎝ℯ ⎠⎠ dx ⎛ 2⎛ x⎞ ⎞ x ⎝tan ⎝ℯ ⎠ + 1⎠⋅ℯ (j) d ⎛ √x⎞ ──⎝x ⎠ dx √x ⎛log(x) 1 ⎞ x ⋅⎜────── + ──⎟ ⎝ 2⋅√x √x⎠ (k) ⎛ ⎛ 6004799503160661⎞⎞ ⎜ ⎜ ─────────────────⎟⎟ ⎜ ⎜ 18014398509481984⎟⎟ d ⎜ ⎝x ⎠⎟ ──⎝x ⎠ dx ⎛ 6004799503160661⎞ ⎜ ─────────────────⎟ ⎜ 18014398509481984⎟ ⎝x ⎠ ⎛ 6004799503160661⋅log(x) 1 x ⋅⎜──────────────────────────────────── + ──────────────── ⎜ 12009599006321323 120095990063213 ⎜ ───────────────── ─────────────── ⎜ 18014398509481984 180143985094819 ⎝18014398509481984⋅x x ⎞ ──⎟ 23⎟ ──⎟ 84⎟ ⎠ (l) d ⎛ ⎛ x⎞⎞ ──⎝asin⎝x + ℯ ⎠⎠ dx x ℯ + 1 ───────────────────── _________________ ╱ 2 ╱ ⎛ x⎞ ╲╱ - ⎝x + ℯ ⎠ + 1 (m) d ⎛ tan(x)⎞ ──⎝ℯ ⎠ dx ⎛ 2 ⎞ tan(x) ⎝tan (x) + 1⎠⋅ℯ (n) d ⎛ ⎛ x⎞⎞ ──⎝atan⎝ℯ ⎠⎠ dx x ℯ ──────── 2⋅x ℯ + 1$


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="0.001" 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="0.1" 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,
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.log(Math.cos(3 * x + 5)),
f1 = (x) => -3 * Math.tan(3 * x + 5)
g = (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),
dx0 = parseFloat(input_dx0.value);

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

let points = [],
lines = [],
fns = [[f, 'red']],
fns1 = [],
fns2 = [[g, 'green']];

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])
let yscale = d3.scaleLinear()
.domain([y1, y2])

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();