$$$\frac{v \left(\ln\left(b\right) + 1\right)}{\ln\left(b\right)}$$$ 對 $$$v$$$ 的導數
相關計算器: 對數微分計算器, 隱式微分計算器（附步驟）
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求$$$\frac{d}{dv} \left(\frac{v \left(\ln\left(b\right) + 1\right)}{\ln\left(b\right)}\right)$$$。
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 $$$c = \frac{\ln\left(b\right) + 1}{\ln\left(b\right)}$$$ 與 $$$f{\left(v \right)} = v$$$：
$${\color{red}\left(\frac{d}{dv} \left(\frac{v \left(\ln\left(b\right) + 1\right)}{\ln\left(b\right)}\right)\right)} = {\color{red}\left(\frac{\ln\left(b\right) + 1}{\ln\left(b\right)} \frac{d}{dv} \left(v\right)\right)}$$套用冪次法則 $$$\frac{d}{dv} \left(v^{n}\right) = n v^{n - 1}$$$，取 $$$n = 1$$$，也就是 $$$\frac{d}{dv} \left(v\right) = 1$$$：
$$\frac{\left(\ln\left(b\right) + 1\right) {\color{red}\left(\frac{d}{dv} \left(v\right)\right)}}{\ln\left(b\right)} = \frac{\left(\ln\left(b\right) + 1\right) {\color{red}\left(1\right)}}{\ln\left(b\right)}$$化簡：
$$\frac{\ln\left(b\right) + 1}{\ln\left(b\right)} = 1 + \frac{1}{\ln\left(b\right)}$$因此，$$$\frac{d}{dv} \left(\frac{v \left(\ln\left(b\right) + 1\right)}{\ln\left(b\right)}\right) = 1 + \frac{1}{\ln\left(b\right)}$$$。
答案
$$$\frac{d}{dv} \left(\frac{v \left(\ln\left(b\right) + 1\right)}{\ln\left(b\right)}\right) = 1 + \frac{1}{\ln\left(b\right)}$$$A