大学数学基礎解説

xの整数、有理数乗の微分の証明(テスト)

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この記事では微分の定義からxの正数、負数、有理数乗の単項式の微分の公式を導出する。
なお、微分の定義は

微分の定義

$f^{'} (x) = lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

とする。

今回証明するもの

xの指数が正数の時の微分
xの指数が負数の時の微分
xの指数が有理数の時の微分

1. xの指数が正数の時の微分

$f (x) := x^{n} (n \in N, n \notin 0)$
$f^{'} (x) = n x^{n - 1}$

$f^{'} (x)$

$= lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

$= lim_{h \to 0} \frac{(x + h)^{n} - x^{n}}{h}$

$= lim_{h \to 0} \frac{(x^{n} +_{n} C_{1} x^{n - 1} h^{} +_{n} C_{2} x^{n - 2} h^{2} + \dots_{n} C_{n - 2} x^{2} h^{n - 2} +_{n} C_{n - 1} x^{} h^{n - 1} +_{n} C_{n} h^{n}) - x^{n}}{h}$

$= lim_{h \to 0} \frac{_{n} C_{1} x^{n - 1} h^{} +_{n} C_{2} x^{n - 2} h^{2} + \dots_{n} C_{n - 2} x^{2} h^{n - 2} +_{n} C_{n - 1} x^{} h^{n - 1} +_{n} C_{n} h^{n}}{h}$

$= lim_{h \to 0} (_{n} C_{1} x^{n - 1} +_{n} C_{2} x^{n - 2} h^{} +_{n} C_{3} x^{n - 3} h^{2} + \dots_{n} C_{n - 2} x^{2} h^{n - 3} +_{n} C_{n - 1} x^{} h^{n - 2} +_{n} C_{n} h^{n - 1})$

$=_{n} C_{1} x^{n - 1} +_{n} C_{2} x^{n - 2} \cdot 0 +_{n} C_{3} x^{n - 3} \cdot 0 + \dots_{n} C_{n - 2} x^{2} \cdot 0 +_{n} C_{n - 1} x^{} \cdot 0 +_{n} C_{n} \cdot 0$

$=_{n} C_{1} x^{n - 1}$

$= n x^{n - 1} ◻$

2. xの指数が正数の時の微分

$f (x) := x^{- n} (n \in N, n \notin 0)$
$f^{'} (x) = - n x^{- n - 1}$

$f^{'} (x)$

$= lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

$= lim_{h \to 0} \frac{(x + h)^{- n} - x^{- n}}{h}$

$= lim_{h \to 0} \frac{\frac{1}{(x + h)^{n}} - \frac{1}{x^{n}}}{h}$

$= lim_{h \to 0} \frac{\frac{x^{n} - (x + h)^{n}}{(x + h)^{n} x^{n}}}{h}$

$= lim_{h \to 0} \frac{x^{n} - (x^{n} +_{n} C_{1} x^{n - 1} h^{} +_{n} C_{2} x^{n - 2} h^{2} + \dots_{n} C_{n - 2} x^{2} h^{n - 2} +_{n} C_{n - 1} x^{} h^{n - 1} +_{n} C_{n} h^{n})}{h (x + h)^{n} x^{n}}$

$= lim_{h \to 0} \frac{- (_{n} C_{1} x^{n - 1} h^{} +_{n} C_{2} x^{n - 2} h^{2} + \dots_{n} C_{n - 2} x^{2} h^{n - 2} +_{n} C_{n - 1} x^{} h^{n - 1} +_{n} C_{n} h^{n})}{h (x + h)^{n} x^{n}}$

$= lim_{h \to 0} \frac{- (_{n} C_{1} x^{n - 1} +_{n} C_{2} x^{n - 2} h^{} +_{n} C_{3} x^{n - 3} h^{2} + \dots_{n} C_{n - 2} x^{2} h^{n - 3} +_{n} C_{n - 1} x^{} h^{n - 2} +_{n} C_{n} h^{n - 1})}{(x + h)^{n} x^{n}}$

$= \frac{- (_{n} C_{1} x^{n - 1} +_{n} C_{2} x^{n - 2} \cdot 0 +_{n} C_{3} x^{n - 3} \cdot 0 + \dots_{n} C_{n - 2} x^{2} \cdot 0 +_{n} C_{n - 1} x^{} \cdot 0 +_{n} C_{n} \cdot 0)}{(x + 0)^{n} x^{n}}$

$= \frac{- n x^{n - 1}}{x^{n} x^{n}}$

$= - n \frac{x^{n - 1}}{x^{2 n}}$

$= - n x^{n - 1 - 2 n}$

$= - n x^{- n - 1} ◻$

3. xの指数が有理数の時の微分

$f (x) := x^{\frac{n}{m}} (n \in Z, m \in N, m \neq 0)$
$f^{'} (x) = \frac{n}{m} x^{\frac{n}{m} - 1}$

これを証明するために $a^{n} - b^{n}$ の展開公式を使う

$a^{n} - b^{n} = (a - b) (a^{n - 1} + a^{n - 2} b + a^{n - 3} b^{2} + \dots + b^{n - 1})$

定理4を使い定理3を証明する

$f^{'} (x)$

$= lim_{h \to 0} \frac{f (x + h) - f (x)}{h}$

$= lim_{h \to 0} \frac{(x + h)^{\frac{n}{m}} - x^{\frac{n}{m}}}{h}$

$= lim_{h \to 0} \frac{\sqrt[m]{(x + h)^{n}} - \sqrt[m]{x^{n}}}{h}$

$= lim_{h \to 0} \frac{(\sqrt[m]{(x + h)} - \sqrt[m]{x}) (\sqrt[m]{(x + h)^{n - 1}} + \sqrt[m]{(x + h)^{n - 2} x^{}} + \sqrt[m]{(x + h)^{n - 3} x^{2}} + \dots + \sqrt[m]{x^{n - 1}})}{h}$

$= lim_{h \to 0} \frac{(\sqrt[m]{(x + h)} - \sqrt[m]{x}) (\sqrt[m]{(x + h)^{m - 1}} + \sqrt[m]{(x + h)^{m - 2} x^{}} + \sqrt[m]{(x + h)^{m - 3} x^{2}} + \dots + \sqrt[m]{x^{m - 1}}) (\sqrt[m]{(x + h)^{n - 1}} + \sqrt[m]{(x + h)^{n - 2} x^{}} + \sqrt[m]{(x + h)^{n - 3} x^{2}} + \dots + \sqrt[m]{x^{n - 1}})}{h (\sqrt[m]{(x + h)^{m - 1}} + \sqrt[m]{(x + h)^{m - 2} x^{}} + \sqrt[m]{(x + h)^{m - 3} x^{2}} + \dots + \sqrt[m]{x^{m - 1}})}$

$= lim_{h \to 0} \frac{(\sqrt[m]{(x + h)^{m}} - \sqrt[m]{x^{m}}) (\sqrt[m]{(x + h)^{n - 1}} + \sqrt[m]{(x + h)^{n - 2} x^{}} + \sqrt[m]{(x + h)^{n - 3} x^{2}} + \dots + \sqrt[m]{x^{n - 1}})}{h (\sqrt[m]{(x + h)^{m - 1}} + \sqrt[m]{(x + h)^{m - 2} x^{}} + \sqrt[m]{(x + h)^{m - 3} x^{2}} + \dots + \sqrt[m]{x^{m - 1}})}$

$= lim_{h \to 0} \frac{(x + h - x) (\sqrt[m]{(x + h)^{n - 1}} + \sqrt[m]{(x + h)^{n - 2} x^{}} + \sqrt[m]{(x + h)^{n - 3} x^{2}} + \dots + \sqrt[m]{x^{n - 1}})}{h (\sqrt[m]{(x + h)^{m - 1}} + \sqrt[m]{(x + h)^{m - 2} x^{}} + \sqrt[m]{(x + h)^{m - 3} x^{2}} + \dots + \sqrt[m]{x^{m - 1}})}$

$= lim_{h \to 0} \frac{h (\sqrt[m]{(x + h)^{n - 1}} + \sqrt[m]{(x + h)^{n - 2} x^{}} + \sqrt[m]{(x + h)^{n - 3} x^{2}} + \dots + \sqrt[m]{x^{n - 1}})}{h (\sqrt[m]{(x + h)^{m - 1}} + \sqrt[m]{(x + h)^{m - 2} x^{}} + \sqrt[m]{(x + h)^{m - 3} x^{2}} + \dots + \sqrt[m]{x^{m - 1}})}$

$= lim_{h \to 0} \frac{\sqrt[m]{(x + h)^{n - 1}} + \sqrt[m]{(x + h)^{n - 2} x^{}} + \sqrt[m]{(x + h)^{n - 3} x^{2}} + \dots + \sqrt[m]{x^{n - 1}}}{\sqrt[m]{(x + h)^{m - 1}} + \sqrt[m]{(x + h)^{m - 2} x^{}} + \sqrt[m]{(x + h)^{m - 3} x^{2}} + \dots + \sqrt[m]{x^{m - 1}}}$

$= \frac{\sqrt[m]{(x + 0)^{n - 1}} + \sqrt[m]{(x + 0)^{n - 2} x^{}} + \sqrt[m]{(x + 0)^{n - 3} x^{2}} + \dots + \sqrt[m]{x^{n - 1}}}{\sqrt[m]{(x + 0)^{m - 1}} + \sqrt[m]{(x + 0)^{m - 2} x^{}} + \sqrt[m]{(x + 0)^{m - 3} x^{2}} + \dots + \sqrt[m]{x^{m - 1}}}$

$= \frac{\sqrt[m]{x^{n - 1}} + \sqrt[m]{x^{n - 2} x^{}} + \sqrt[m]{x^{n - 3} x^{2}} + \dots + \sqrt[m]{x^{n - 1}}}{\sqrt[m]{x^{m - 1}} + \sqrt[m]{x^{m - 2} x^{}} + \sqrt[m]{x^{m - 3} x^{2}} + \dots + \sqrt[m]{x^{m - 1}}}$

$= \frac{\sqrt[m]{x^{n - 1}} + \sqrt[m]{x^{n - 1}} + \sqrt[m]{x^{n - 1}} + \dots + \sqrt[m]{x^{n - 1}}}{\sqrt[m]{x^{m - 1}} + \sqrt[m]{x^{m - 1}} + \sqrt[m]{x^{m - 1}} + \dots + \sqrt[m]{x^{m - 1}}}$