数学 - Python - 微分積分学 - 微分法の公式 - 導関数の求め方 - 累乗、累乗根、有理式

学習環境

微分積分学 (ちくま学芸文庫) (吉田洋一(著)、筑摩書房)のⅡ.(微分法の公式)、6.(導関数の求め方)、演習問題II、1、2、3、4、5、6、7、8、9、10、11、12、13、14、15の解答を求めてみる。

$\begin{array}{l} \frac{d}{dx} (x^{5} + 4 x^{3} + 7 x + 3) \\ = 5 x^{4} + 12 x^{2} + 7 \end{array}$
$\begin{array}{l} \frac{d}{dx} \frac{a + b x + c x^{2}}{\sqrt{x}} \\ = \frac{(b + 2 c x) \sqrt{x} - (a + b x + c x^{2}) \frac{1}{2 \sqrt{x}}}{x} \\ = \frac{2 x (b + 2 c x) - (a + b x + c x^{2})}{2 x \sqrt{x}} \\ = \frac{3 c x^{2} + b x - a}{2 x \sqrt{x}} \end{array}$
$\begin{array}{l} \frac{d}{dx} \frac{3 x + 2}{1 + x + x^{2}} \\ = \frac{3 (1 + x + x^{2}) - (3 x + 2) (1 + 2 x)}{{(1 + x + x^{2})}^{2}} \\ = \frac{1 - 4 x - 3 x^{2}}{{(1 + x + x^{2})}^{2}} \end{array}$
$\begin{array}{l} \frac{d}{dx} \frac{x - a}{(x - b) (x - c)} \\ = \frac{(x - b) (x - c) - (x - a) (x - c + x - b)}{{(x - b)}^{2} {(x - c)}^{2}} \\ = \frac{- x^{2} + (- b - c + b + c + 2 a) x + b c - a b - a c}{{(x - b)}^{2} {(x - c)}^{2}} \\ = \frac{- x^{2} + 2 a x - a b + b c - c a}{{(x - b)}^{2} {(x - c)}^{2}} \end{array}$
$\begin{array}{l} \frac{d}{dx} (x (x + a) (x + b)) \\ = (x + a) (x + b) + x (x + b) + x (x + a) \\ = 3 x^{2} + 2 (a + b) x + a b \end{array}$
$\begin{array}{l} \frac{d}{dx} \frac{1}{{({(x - a)}^{2} + b^{2})}^{n}} \\ = - \frac{n {({(x - a)}^{2} + b^{2})}^{n - 1} 2 (x - a)}{{({(x - a)}^{2} + b^{2})}^{2 n}} \end{array}$
$\begin{array}{l} \frac{d}{dx} \sqrt{a^{4} - x^{4}} \\ = \frac{1}{2 \sqrt{a^{4} - x^{4}}} \cdot (- 4 x^{3}) \\ = - \frac{2 x^{3}}{\sqrt{a^{4} - x^{4}}} \end{array}$
$\begin{array}{l} \frac{d}{dx} \sqrt{(x - 2) (x - 3)} \\ = \frac{1}{2 \sqrt{(x - 2) (x - 3)}} \cdot (x - 3 + x - 2) \\ = \frac{2 x - 5}{2 \sqrt{(x - 2) (x - 3)}} \end{array}$
$\begin{array}{l} \frac{d}{dx} \frac{\sqrt{1 + 2 x}}{\sqrt[3]{1 + 3 x}} \\ = \frac{d}{dx} ({(1 + 2 x)}^{\frac{1}{2}} {(1 + 3 x)}^{- \frac{1}{3}}) \\ = \frac{1}{2} {(1 + 2 x)}^{- \frac{1}{2}} 2 {(1 + 3 x)}^{- \frac{1}{3}} + {(1 + 2 x)}^{\frac{1}{2}} \cdot (- \frac{1}{3}) {(1 + 3 x)}^{- \frac{4}{3}} \cdot 3 \\ = {(1 + 2 x)}^{- \frac{1}{2}} {(1 + 3 x)}^{- \frac{1}{3}} - {(1 + 2 x)}^{\frac{1}{2}} {(1 + 3 x)}^{- \frac{4}{3}} \end{array}$
$\begin{array}{l} \frac{d}{dx} \sqrt{\frac{1 - \sqrt[3]{x}}{1 + \sqrt[3]{x}}} \\ = \frac{1}{2} \sqrt{\frac{1 + \sqrt[3]{x}}{1 - \sqrt[3]{x}}} \cdot \frac{1}{{(1 + \sqrt[3]{x})}^{2}} (- \frac{1}{3} x^{- \frac{2}{3}} (1 + \sqrt[3]{x}) - (1 - \sqrt[3]{x}) \cdot \frac{1}{3} x^{- \frac{2}{3}}) \\ = - \frac{x^{- \frac{2}{3}}}{3} \sqrt{\frac{1 + \sqrt[3]{x}}{1 - \sqrt[3]{x}}} \frac{1}{{(1 + \sqrt[3]{x})}^{2}} \\ = - \frac{1}{3} x^{- \frac{2}{3}} \frac{1}{\sqrt{(1 - \sqrt[3]{x}) (1 + \sqrt[3]{x})}} \cdot \frac{1}{1 + \sqrt[3]{x}} \\ = - \frac{1}{3} \frac{1}{x^{\frac{2}{3}} \sqrt{1 - \sqrt[3]{x^{2}}}} \cdot \frac{1}{1 + \sqrt[3]{x}} \end{array}$
$\begin{array}{l} \frac{d}{dx} \frac{\sqrt{a^{2} + x^{2}} + \sqrt{a^{2} - x^{2}}}{\sqrt{a^{2} + x^{2}} - \sqrt{a^{2} - x^{2}}} \\ = \frac{d}{dx} \frac{a^{2} + x^{2} + 2 \sqrt{a^{4} - x^{4}} + a^{2} - x^{2}}{(a^{2} + x^{2}) - (a^{2} - x^{2})} \\ = \frac{d}{dx} \frac{2 a^{2} + 2 \sqrt{a^{4} - x^{4}}}{2 x^{2}} \\ = \frac{d}{dx} \frac{a^{2} + \sqrt{a^{4} - x^{4}}}{x^{2}} \\ = \frac{\frac{- 4 x^{3}}{2 \sqrt{a^{4} - x^{4}}} x^{2} - (a^{2} + \sqrt{a^{4} - x^{4}}) 2 x}{x^{4}} \\ = \frac{- \frac{2 x^{4}}{\sqrt{a^{4} - x^{4}}} - (a^{2} + \sqrt{a^{4} - x^{4}}) 2}{x^{3} \sqrt{a^{4} - x^{4}}} \\ = \frac{- 2 x^{4} - (a^{2} + \sqrt{a^{4} - x^{4}}) 2 \sqrt{a^{4} - x^{4}}}{x^{3} \sqrt{a^{4} - x^{4}}} \\ = \frac{- 2 (x^{4} + a^{2} \sqrt{a^{4} - x^{4}} + a^{4} - x^{4})}{x^{3} \sqrt{a^{4} - x^{4}}} \\ = \frac{- 2 a^{2} (\sqrt{a^{4} - x^{4}} + a^{2})}{x^{3} \sqrt{a^{4} - x^{4}}} \end{array}$
$\begin{array}{l} \frac{d}{dx} {(a^{\frac{2}{3}} - x^{\frac{2}{3}})}^{\frac{3}{2}} \\ = \frac{3}{2} {(a^{\frac{2}{3}} - x^{\frac{2}{3}})}^{\frac{1}{2}} \cdot (- \frac{2}{3} x^{- \frac{1}{3}}) \\ = - {(a^{\frac{2}{3}} - x^{\frac{2}{3}})}^{\frac{1}{2}} x^{- \frac{1}{3}} \end{array}$
$\begin{array}{l} \frac{d}{dx} \frac{1}{{(x + 1)}^{m} {(x + 3)}^{n}} \\ = \frac{d}{dx} {(x + 1)}^{- m} {(x + 3)}^{- n} \\ = - m {(x + 1)}^{- (m - 1)} {(x + 3)}^{- n} + {(x + 1)}^{- m} (- n) {(x + 3)}^{- (n - 1)} \end{array}$
$\begin{array}{l} \frac{d}{dx} x^{m} {(a + b x^{n})}^{p} \\ = m x^{m - 1} {(a + b x^{n})}^{p} + x^{m} p {(a + b x^{n})}^{p - 1} b x^{n - 1} \end{array}$
$\begin{array}{l} \frac{d}{dx} \frac{1}{x + \sqrt{x^{2} + a^{2}}} \\ = - \frac{1 + \frac{2 x}{2 \sqrt{x^{2} + a^{2}}}}{{(x + \sqrt{x^{2} + a^{2}})}^{2}} \\ = - \frac{\sqrt{x^{2} + a^{2}} + x}{{(x + \sqrt{x^{2} + a^{2}})}^{2} \sqrt{x^{2} + a^{2}}} \\ = - \frac{1}{\sqrt{(x + \sqrt{x^{2} + a^{2}}) \sqrt[2]{x^{2} + a^{2}}}} \end{array}$

#!/usr/bin/env python3 from sympy import pprint, symbols, sqrt, root, Derivative, Rational from unittest import TestCase, main print('1-15.') class MyTest(TestCase): def setUp(self): pass def tearDown(self): pass def test(self): x, a, b, c, m, n, p = symbols('x, a, b, c, m, n, p') spam = [x ** 5 + 4 * x ** 3 + 7 * x + 3, (a + b * x + c * x ** 2) / sqrt(x), (3 * x + 2) / (1 + x + x ** 2), (x - a) / ((x - b) * (x - c)), x * (x + a) * (x + b), 1 / ((x - a) ** 2 + b ** 2) ** n, sqrt(a ** 4 - x ** 4), sqrt((x - 2) * (x - 3)), sqrt(1 + 2 * x) / root(1 + 3 * x, 3), sqrt((1 - root(x, 3)) / (1 + root(x, 3))), (sqrt(a ** 2 + x ** 2) + (a ** 2 - x ** 2)) / (sqrt(a ** 2 + x ** 2) - sqrt(a ** 2 - x ** 2)), (a ** Rational(2, 3) - x ** Rational(2, 3)) ** Rational(3, 2), 1 / ((x + 1) ** m * (x + 3) ** n), x ** m * (a + b * x ** n) ** p, 1 / (x + sqrt(x ** 2 + a ** 2))] egg = [5 * x ** 4 + 12 * x ** 2 + 7, (3 * c * x ** 2 + b * x - a) / (2 * x * sqrt(x)), (1 - 4 * x - 3 * x ** 2) / (1 + x + x ** 2) ** 2, (-x ** 2 + 2 * a * x - a * b + b * c - c * a) / ((x - b) ** 2 * (x - c) ** 2), 3 * x ** 2 + 2 * (a + b) * x + a * b, - (n * ((x - a) ** 2 + b ** 2) ** (n - 1) * 2 * (x - a)) / ((x - a) ** 2 + b ** 2) ** (2 * n), -2 * x ** 3 / sqrt(a ** 4 - x ** 4), (2 * x - 5) / (2 * sqrt((x - 2) * (x - 3))), (1 + 2 * x) ** -Rational(1, 2) * (1 + 3 * x) ** -Rational(1, 3) - (1 + 2 * x) ** Rational(1, 2) * (1 + 3 * x) ** -Rational(4, 3), -Rational(1, 3) * x ** -Rational(2, 3) * 1 / sqrt((1 - root(x, 3)) * (1 + root(x, 3))) * 1 / (1 + root(x, 3)), -2 * a ** 2 * (sqrt(a ** 4 - x ** 4) + a ** 2) / (x ** 3 * sqrt(a ** 4 - x ** 4)), -(a ** Rational(2, 3) - x ** Rational(2, 3)) ** Rational(1, 2) * x ** -Rational(1, 3), -m * (x + 1) ** -(m - 1) * (x + 3) ** -n + (x + 1) ** -m * (-n) * (x + 3) ** -(n - 1), m * x ** (m - 1) * (a + b * x ** n) ** p + x ** m * p * (a + b * x ** n) ** (p - 1) * b * x ** (n - 1), -1 / ((x + sqrt(x ** 2 + a ** 2)) * sqrt(x ** 2 + a ** 2))] for i, (s, t) in enumerate(zip(spam, egg), 1): d = Derivative(s, x, 1).doit() try: self.assertEqual(d.factor(), t.factor()) except: try: self.assertEqual(d.simplify(), t.simplify()) except: try: self.assertEqual(d.expand(), t.expand()) except AssertionError as err: for o in [f'{i}.', s.simplify(), d.factor(), t]: pprint(o) print() print() if __name__ == '__main__': main()

入出力結果(Bash、cmd.exe(コマンドプロンプト)、Terminal、Jupyter(IPython))

$ ./sample1.py 1-15. 10. ___________ ╱ 3 ___ ╱ 1 - ╲╱ x ╱ ───────── ╱ 3 ___ ╲╱ ╲╱ x + 1 _______________ ╱ ⎛3 ___ ⎞ ╱ -⎝╲╱ x - 1⎠ ╱ ───────────── ╱ 3 ___ ╲╱ ╲╱ x + 1 ────────────────────────────── 2/3 ⎛3 ___ ⎞ ⎛3 ___ ⎞ 3⋅x ⋅⎝╲╱ x - 1⎠⋅⎝╲╱ x + 1⎠ -1 ─────────────────────────────────────────────── _________________________ 2/3 ╱ ⎛ 3 ___⎞ ⎛3 ___ ⎞ ⎛3 ___ ⎞ 3⋅x ⋅╲╱ ⎝1 - ╲╱ x ⎠⋅⎝╲╱ x + 1⎠ ⋅⎝╲╱ x + 1⎠ 11. _________ 2 2 ╱ 2 2 - a + x - ╲╱ a + x ─────────────────────────── _________ _________ ╱ 2 2 ╱ 2 2 ╲╱ a - x - ╲╱ a + x ⎛ _________ _________ _________ _____ ⎜ 2 ╱ 2 2 2 ╱ 2 2 2 2 ╱ 2 2 2 ╱ 2 -x⋅⎝3⋅a ⋅╲╱ a - x - a ⋅╲╱ a + x + 2⋅a + x ⋅╲╱ a - x + x ⋅╲╱ a + ────────────────────────────────────────────────────────────────────────────── 2 _________ ⎛ _________ _________⎞ ___________________ ╱ 2 2 ⎜ ╱ 2 2 ╱ 2 2 ⎟ ╲╱ -(-a + x)⋅(a + x) ⋅╲╱ a + x ⋅⎝- ╲╱ a - x + ╲╱ a + x ⎠ ____⎞ 2 ⎟ x ⎠ ────── ⎛ _________⎞ 2 ⎜ 2 ╱ 4 4 ⎟ -2⋅a ⋅⎝a + ╲╱ a - x ⎠ ────────────────────────── _________ 3 ╱ 4 4 x ⋅╲╱ a - x 13. -m -n (x + 1) ⋅(x + 3) -m -n -(x + 1) ⋅(x + 3) ⋅(m⋅x + 3⋅m + n⋅x + n) ─────────────────────────────────────────── (x + 1)⋅(x + 3) 1 - m -n -m 1 - n - m⋅(x + 1) ⋅(x + 3) - n⋅(x + 1) ⋅(x + 3) 14. p m ⎛ n⎞ x ⋅⎝a + b⋅x ⎠ p m ⎛ n⎞ ⎛ n n⎞ x ⋅⎝a + b⋅x ⎠ ⋅⎝a⋅m + b⋅m⋅x + b⋅n⋅p⋅x ⎠ ──────────────────────────────────────── ⎛ n⎞ x⋅⎝a + b⋅x ⎠ p - 1 p m n - 1 ⎛ n⎞ m - 1 ⎛ n⎞ b⋅p⋅x ⋅x ⋅⎝a + b⋅x ⎠ + m⋅x ⋅⎝a + b⋅x ⎠ . ---------------------------------------------------------------------- Ran 1 test in 2.027s OK $

計算間違いしてる可能性があるけど、気にせず先に進めていくことに。

Kamimura's blog

2019年10月1日火曜日

数学 - Python - 微分積分学 - 微分法の公式 - 導関数の求め方 - 累乗、累乗根、有理式

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