2*atan(v) 的導數

此計算器將求出 $$$2 \operatorname{atan}{\left(v \right)}$$$ 的導數，並顯示步驟。

您的輸入

求$$$\frac{d}{dv} \left(2 \operatorname{atan}{\left(v \right)}\right)$$$。

解答

套用常數倍法則 $$$\frac{d}{dv} \left(c f{\left(v \right)}\right) = c \frac{d}{dv} \left(f{\left(v \right)}\right)$$$，使用 $$$c = 2$$$ 與 $$$f{\left(v \right)} = \operatorname{atan}{\left(v \right)}$$$：

$${\color{red}\left(\frac{d}{dv} \left(2 \operatorname{atan}{\left(v \right)}\right)\right)} = {\color{red}\left(2 \frac{d}{dv} \left(\operatorname{atan}{\left(v \right)}\right)\right)}$$

反正切函數的導數是 $$$\frac{d}{dv} \left(\operatorname{atan}{\left(v \right)}\right) = \frac{1}{v^{2} + 1}$$$:

$$2 {\color{red}\left(\frac{d}{dv} \left(\operatorname{atan}{\left(v \right)}\right)\right)} = 2 {\color{red}\left(\frac{1}{v^{2} + 1}\right)}$$

因此，$$$\frac{d}{dv} \left(2 \operatorname{atan}{\left(v \right)}\right) = \frac{2}{v^{2} + 1}$$$。

答案

$$$\frac{d}{dv} \left(2 \operatorname{atan}{\left(v \right)}\right) = \frac{2}{v^{2} + 1}$$$A

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