$$$v$$$에 대한 $$$\frac{v \left(\ln\left(b\right) + 1\right)}{\ln\left(b\right)}$$$의 도함수
관련 계산기: 로그 미분 계산기, 암시적 미분 계산기 (단계별 풀이)
사용자 입력
$$$\frac{d}{dv} \left(\frac{v \left(\ln\left(b\right) + 1\right)}{\ln\left(b\right)}\right)$$$을(를) 구하시오.
풀이
상수배 법칙 $$$\frac{d}{dv} \left(c f{\left(v \right)}\right) = c \frac{d}{dv} \left(f{\left(v \right)}\right)$$$을 $$$c = \frac{\ln\left(b\right) + 1}{\ln\left(b\right)}$$$와 $$$f{\left(v \right)} = v$$$에 적용합니다:
$${\color{red}\left(\frac{d}{dv} \left(\frac{v \left(\ln\left(b\right) + 1\right)}{\ln\left(b\right)}\right)\right)} = {\color{red}\left(\frac{\ln\left(b\right) + 1}{\ln\left(b\right)} \frac{d}{dv} \left(v\right)\right)}$$멱법칙 $$$\frac{d}{dv} \left(v^{n}\right) = n v^{n - 1}$$$을 $$$n = 1$$$에 대해 적용하면, 즉 $$$\frac{d}{dv} \left(v\right) = 1$$$:
$$\frac{\left(\ln\left(b\right) + 1\right) {\color{red}\left(\frac{d}{dv} \left(v\right)\right)}}{\ln\left(b\right)} = \frac{\left(\ln\left(b\right) + 1\right) {\color{red}\left(1\right)}}{\ln\left(b\right)}$$간단히 하시오:
$$\frac{\ln\left(b\right) + 1}{\ln\left(b\right)} = 1 + \frac{1}{\ln\left(b\right)}$$따라서, $$$\frac{d}{dv} \left(\frac{v \left(\ln\left(b\right) + 1\right)}{\ln\left(b\right)}\right) = 1 + \frac{1}{\ln\left(b\right)}$$$.
정답
$$$\frac{d}{dv} \left(\frac{v \left(\ln\left(b\right) + 1\right)}{\ln\left(b\right)}\right) = 1 + \frac{1}{\ln\left(b\right)}$$$A