数学 - Python - 微分積分学 - 微分法の公式 - 微分法の公式 - 合成関数の導関数

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)、1.(微分法の公式)、問3の解答を求めてみる。

1. $\begin{array}{l} \frac{d}{dx} \sqrt{2 x - \frac{1}{x^{2}}} \\ = \frac{1}{2 \sqrt{2 x - \frac{1}{x^{2}}}} \cdot (2 + \frac{2 x}{x^{4}}) \\ = \frac{1}{\sqrt{2 x - \frac{1}{x^{2}}}} (1 + \frac{1}{x^{3}}) \\ = \frac{x^{3} + 1}{x^{2} \sqrt{2 x^{3} - 1}} \end{array}$
2. $\begin{array}{l} \frac{d}{dx} \sqrt{x + \sqrt{1 + x^{2}}} \\ = \frac{1}{2 \sqrt{x + \sqrt{1 + x^{2}}}} \cdot (1 + \frac{1}{2 \sqrt{1 + x^{2}}} \cdot 2 x) \\ = \frac{1}{2 \sqrt{x + \sqrt{1 + x^{2}}}} \cdot \frac{\sqrt{1 + x^{2}} + x}{\sqrt{1 + x^{2}}} \\ = \frac{\sqrt{x + \sqrt{1 + x^{2}}}}{2 \sqrt{1 + x^{2}}} \end{array}$
3. $\begin{array}{l} \frac{d}{dx} \sqrt{\frac{(a + x) (b + x)}{(a - x) (b - x)}} \\ = \frac{1}{2} \sqrt{\frac{(a - x) (b - x)}{(a + x) (b + x)}} \cdot \\ \frac{(b + x + a + x) (a - x) (b - x) - (a + x) (b + x) (- (b - x) - (a - x))}{{(a - x)}^{2} {(b - x)}^{2}} \\ = \frac{1}{2} \sqrt{\frac{(a - x) (b - x)}{(a + x) (b + x)}} \cdot \frac{(a + b + 2 x) (a - x) (b - x) - (a + x) (b + x) (- a - b + 2 x)}{{(a - x)}^{2} {(b - x)}^{2}} \\ = \frac{1}{2} \frac{(a - x) (b - x)}{\sqrt{(a^{2} - x^{2}) (b^{2} - x^{2})}} \cdot (\frac{(a + b) ((a - x) (b - x) + (a + x) (b + x))}{{(a - x)}^{2} {(b - x)}^{2}} + \\ \frac{2 x ((a - x) (b - x) - (a + x) (b + x))}{{(a - x)}^{2} {(b - x)}^{2}}) \\ = \frac{1}{2} \cdot \frac{1}{\sqrt{(a^{2} - x^{2}) (b^{2} - x^{2})}} \cdot \frac{1}{(a - x) (b - x)} ((a + b) (2 a b + 2 x^{2}) + 2 x (- 2 a x - 2 b x)) \\ = \frac{(a + b) (a b + x^{2} - 2 x^{2})}{\sqrt{(a^{2} - x^{2}) (b^{2} - x^{2})} (a - x) (b - x)} \\ = \frac{(a + b) (a b - x^{2})}{\sqrt{(a^{2} - x^{2}) (b^{2} - x^{2})} (a - x) (b - x)} \end{array}$

#!/usr/bin/env python3 from sympy import pprint, symbols, sqrt, Derivative print('3.') x, a, b = symbols('x, a, b') fs = [sqrt(2 * x - 1 / x ** 2), sqrt(x + sqrt(1 + x ** 2)), sqrt((a + x) * (b + x) / ((a - x) * (b - x)))] dfs1 = [Derivative(f, x, 1).doit() for f in fs] dfs2 = [(x ** 3 + 1) / (x ** 2 * sqrt(2 * x ** 3 - 1)), sqrt(x + sqrt(1 + x ** 2)) / (2 * sqrt(1 + x ** 2)), (a + b) * (a * b - x ** 2) / (sqrt((a ** 2 - x ** 2) * (b ** 2 - x ** 2)) * (a - x) * (b - x))] for i, o in enumerate(zip(dfs1, dfs2), 1): print(f'({i})') for t in o: pprint(t.factor()) print()

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

$ ./sample3.py 3. (1) ⎛ 2 ⎞ (x + 1)⋅⎝x - x + 1⎠ ──────────────────── __________ 3 ╱ 1 x ⋅ ╱ 2⋅x - ── ╱ 2 ╲╱ x ⎛ 2 ⎞ (x + 1)⋅⎝x - x + 1⎠ ──────────────────── __________ 2 ╱ 3 x ⋅╲╱ 2⋅x - 1 (2) _________________ ╱ ________ ╱ ╱ 2 ╲╱ x + ╲╱ x + 1 ───────────────────── ________ ╱ 2 2⋅╲╱ x + 1 _________________ ╱ ________ ╱ ╱ 2 ╲╱ x + ╲╱ x + 1 ───────────────────── ________ ╱ 2 2⋅╲╱ x + 1 (3) ______________________ ╱ (a + x)⋅(b + x) ⎛ 2⎞ - ╱ ──────────────────── ⋅(a + b)⋅⎝-a⋅b + x ⎠ ╱ 2 ╲╱ a⋅b - a⋅x - b⋅x + x ───────────────────────────────────────────────── (-a + x)⋅(a + x)⋅(-b + x)⋅(b + x) ⎛ 2⎞ -(a + b)⋅⎝-a⋅b + x ⎠ ─────────────────────────────────────────────────────── ___________________________________ ╲╱ (-a + x)⋅(a + x)⋅(-b + x)⋅(b + x) ⋅(-a + x)⋅(-b + x) $

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2019年9月22日日曜日

数学 - Python - 微分積分学 - 微分法の公式 - 微分法の公式 - 合成関数の導関数

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