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$$$\operatorname{atan}{\left(4 x \right)}$$$の導関数
関連する計算機: 対数微分計算機, 陰関数微分計算機（手順付き）
入力内容
$$$\frac{d}{dx} \left(\operatorname{atan}{\left(4 x \right)}\right)$$$ を求めよ。
解答
関数$$$\operatorname{atan}{\left(4 x \right)}$$$は、2つの関数$$$f{\left(u \right)} = \operatorname{atan}{\left(u \right)}$$$と$$$g{\left(x \right)} = 4 x$$$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:
$${\color{red}\left(\frac{d}{dx} \left(\operatorname{atan}{\left(4 x \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\operatorname{atan}{\left(u \right)}\right) \frac{d}{dx} \left(4 x\right)\right)}$$逆正接関数の導関数は$$$\frac{d}{du} \left(\operatorname{atan}{\left(u \right)}\right) = \frac{1}{u^{2} + 1}$$$:
$${\color{red}\left(\frac{d}{du} \left(\operatorname{atan}{\left(u \right)}\right)\right)} \frac{d}{dx} \left(4 x\right) = {\color{red}\left(\frac{1}{u^{2} + 1}\right)} \frac{d}{dx} \left(4 x\right)$$元の変数に戻す:
$$\frac{\frac{d}{dx} \left(4 x\right)}{{\color{red}\left(u\right)}^{2} + 1} = \frac{\frac{d}{dx} \left(4 x\right)}{{\color{red}\left(4 x\right)}^{2} + 1}$$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 $$$c = 4$$$ と $$$f{\left(x \right)} = x$$$ に対して適用します:
$$\frac{{\color{red}\left(\frac{d}{dx} \left(4 x\right)\right)}}{16 x^{2} + 1} = \frac{{\color{red}\left(4 \frac{d}{dx} \left(x\right)\right)}}{16 x^{2} + 1}$$$$$n = 1$$$ を用いて冪法則 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ を適用すると、すなわち $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$\frac{4 {\color{red}\left(\frac{d}{dx} \left(x\right)\right)}}{16 x^{2} + 1} = \frac{4 {\color{red}\left(1\right)}}{16 x^{2} + 1}$$したがって、$$$\frac{d}{dx} \left(\operatorname{atan}{\left(4 x \right)}\right) = \frac{4}{16 x^{2} + 1}$$$。
解答
$$$\frac{d}{dx} \left(\operatorname{atan}{\left(4 x \right)}\right) = \frac{4}{16 x^{2} + 1}$$$A