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$$$\ln\left(x^{3}\right)$$$の導関数
入力内容
$$$\frac{d}{dx} \left(\ln\left(x^{3}\right)\right)$$$ を求めよ。
解答
関数$$$\ln\left(x^{3}\right)$$$は、2つの関数$$$f{\left(u \right)} = \ln\left(u\right)$$$と$$$g{\left(x \right)} = x^{3}$$$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:
$$\frac{d}{dx} \left(3 \ln\left(x\right)\right) = \frac{d}{dx} \left(3 \ln\left(x\right)\right)$$自然対数の導関数は $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:
$$\frac{d}{dx} \left(3 \ln\left(x\right)\right) = \frac{d}{dx} \left(3 \ln\left(x\right)\right)$$元の変数に戻す:
$$\frac{d}{dx} \left(3 \ln\left(x\right)\right) = \frac{d}{dx} \left(3 \ln\left(x\right)\right)$$定数倍の法則 $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ を $$$c = 3$$$ と $$$f{\left(x \right)} = \ln\left(x\right)$$$ に対して適用します:
$${\color{red}\left(\frac{d}{dx} \left(3 \ln\left(x\right)\right)\right)} = {\color{red}\left(3 \frac{d}{dx} \left(\ln\left(x\right)\right)\right)}$$自然対数の導関数は $$$\frac{d}{dx} \left(\ln\left(x\right)\right) = \frac{1}{x}$$$:
$$3 {\color{red}\left(\frac{d}{dx} \left(\ln\left(x\right)\right)\right)} = 3 {\color{red}\left(\frac{1}{x}\right)}$$したがって、$$$\frac{d}{dx} \left(\ln\left(x^{3}\right)\right) = \frac{3}{x}$$$。
解答
$$$\frac{d}{dx} \left(\ln\left(x^{3}\right)\right) = \frac{3}{x}$$$A