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$$$3^{x}$$$ 的二階導數
您的輸入
求$$$\frac{d^{2}}{dx^{2}} \left(3^{x}\right)$$$。
解答
求第一階導數 $$$\frac{d}{dx} \left(3^{x}\right)$$$
套用指數法則 $$$\frac{d}{dx} \left(n^{x}\right) = n^{x} \ln\left(n\right)$$$，令 $$$n = 3$$$：
$${\color{red}\left(\frac{d}{dx} \left(3^{x}\right)\right)} = {\color{red}\left(3^{x} \ln\left(3\right)\right)}$$因此，$$$\frac{d}{dx} \left(3^{x}\right) = 3^{x} \ln\left(3\right)$$$。
接下來，$$$\frac{d^{2}}{dx^{2}} \left(3^{x}\right) = \frac{d}{dx} \left(3^{x} \ln\left(3\right)\right)$$$
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 $$$c = \ln\left(3\right)$$$ 與 $$$f{\left(x \right)} = 3^{x}$$$：
$${\color{red}\left(\frac{d}{dx} \left(3^{x} \ln\left(3\right)\right)\right)} = {\color{red}\left(\ln\left(3\right) \frac{d}{dx} \left(3^{x}\right)\right)}$$套用指數法則 $$$\frac{d}{dx} \left(n^{x}\right) = n^{x} \ln\left(n\right)$$$，令 $$$n = 3$$$：
$$\ln\left(3\right) {\color{red}\left(\frac{d}{dx} \left(3^{x}\right)\right)} = \ln\left(3\right) {\color{red}\left(3^{x} \ln\left(3\right)\right)}$$因此，$$$\frac{d}{dx} \left(3^{x} \ln\left(3\right)\right) = 3^{x} \ln^{2}\left(3\right)$$$。
因此，$$$\frac{d^{2}}{dx^{2}} \left(3^{x}\right) = 3^{x} \ln^{2}\left(3\right)$$$。
答案
$$$\frac{d^{2}}{dx^{2}} \left(3^{x}\right) = 3^{x} \ln^{2}\left(3\right)$$$A
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