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$$$\ln\left(x^{2}\right)$$$の導関数
入力内容
$$$\frac{d}{dx} \left(\ln\left(x^{2}\right)\right)$$$ を求めよ。
解答
関数$$$\ln\left(x^{2}\right)$$$は、2つの関数$$$f{\left(u \right)} = \ln\left(u\right)$$$と$$$g{\left(x \right)} = x^{2}$$$の合成$$$f{\left(g{\left(x \right)} \right)}$$$である。
連鎖律 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ を適用する:
$${\color{red}\left(\frac{d}{dx} \left(\ln\left(x^{2}\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(x^{2}\right)\right)}$$自然対数の導関数は $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:
$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(x^{2}\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(x^{2}\right)$$元の変数に戻す:
$$\frac{\frac{d}{dx} \left(x^{2}\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dx} \left(x^{2}\right)}{{\color{red}\left(x^{2}\right)}}$$冪法則 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ を $$$n = 2$$$ に対して適用する:
$$\frac{{\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)}}{x^{2}} = \frac{{\color{red}\left(2 x\right)}}{x^{2}}$$したがって、$$$\frac{d}{dx} \left(\ln\left(x^{2}\right)\right) = \frac{2}{x}$$$。
解答
$$$\frac{d}{dx} \left(\ln\left(x^{2}\right)\right) = \frac{2}{x}$$$A
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