$$$x \sqrt{- \ln\left(3\right) + 2 \ln\left(2\right)}$$$ 的導數
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您的輸入
求$$$\frac{d}{dx} \left(x \sqrt{- \ln\left(3\right) + 2 \ln\left(2\right)}\right)$$$。
解答
套用常數倍法則 $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$，使用 $$$c = \sqrt{- \ln\left(3\right) + 2 \ln\left(2\right)}$$$ 與 $$$f{\left(x \right)} = x$$$：
$${\color{red}\left(\frac{d}{dx} \left(x \sqrt{- \ln\left(3\right) + 2 \ln\left(2\right)}\right)\right)} = {\color{red}\left(\sqrt{- \ln\left(3\right) + 2 \ln\left(2\right)} \frac{d}{dx} \left(x\right)\right)}$$套用冪次法則 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$，取 $$$n = 1$$$，也就是 $$$\frac{d}{dx} \left(x\right) = 1$$$：
$$\sqrt{- \ln\left(3\right) + 2 \ln\left(2\right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = \sqrt{- \ln\left(3\right) + 2 \ln\left(2\right)} {\color{red}\left(1\right)}$$化簡：
$$\sqrt{- \ln\left(3\right) + 2 \ln\left(2\right)} = \sqrt{\ln\left(\frac{4}{3}\right)}$$因此，$$$\frac{d}{dx} \left(x \sqrt{- \ln\left(3\right) + 2 \ln\left(2\right)}\right) = \sqrt{\ln\left(\frac{4}{3}\right)}$$$。
答案
$$$\frac{d}{dx} \left(x \sqrt{- \ln\left(3\right) + 2 \ln\left(2\right)}\right) = \sqrt{\ln\left(\frac{4}{3}\right)}$$$A