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実数乗の導関数の問題の解法 | 夢を叶える塾
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実数乗の導関数の問題の解法

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実数乗の導関数を求める問題です。

基本問題

次の関数を微分しなさい。
y=\sqrt{2x+1}

解き方

x^{\alpha}=\alpha x^{\alpha -1}が実数で成り立ちます。

解説

y=\sqrt{2x+1}
\displaystyle y=(2x+1)^{\frac{1}{2}}ですね。
合成関数の微分法を使ってu=2x+1と思って計算を進めましょう。
このときu'=2を忘れないでください。
\displaystyle y'=((2x+1)^{\frac{1}{2}})'=\frac{1}{2}(2x+1)^{\frac{1}{2}-1}\cdot 2
\displaystyle =(2x+1)^{-\frac{1}{2}}=\frac{1}{\sqrt{2x+1}}=\frac{\sqrt{2x+1}}{2x+1}
最後の答えの形は特に指定が無ければ\displaystyle (2x+1)^{-\frac{1}{2}},\frac{1}{\sqrt{2x+1}}でも良いでしょう。

終わりに

元々数Ⅱで自然数のときを扱っていた分、それほど抵抗なく覚えることができると思います。

補足メモ

問題検討中です。

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