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$$$\ln\left(2 u\right)$$$의 도함수
사용자 입력
$$$\frac{d}{du} \left(\ln\left(2 u\right)\right)$$$을(를) 구하시오.
풀이
함수 $$$\ln\left(2 u\right)$$$는 두 함수 $$$f{\left(v \right)} = \ln\left(v\right)$$$와 $$$g{\left(u \right)} = 2 u$$$의 합성함수 $$$f{\left(g{\left(u \right)} \right)}$$$이다.
연쇄법칙 $$$\frac{d}{du} \left(f{\left(g{\left(u \right)} \right)}\right) = \frac{d}{dv} \left(f{\left(v \right)}\right) \frac{d}{du} \left(g{\left(u \right)}\right)$$$을(를) 적용하십시오:
$${\color{red}\left(\frac{d}{du} \left(\ln\left(2 u\right)\right)\right)} = {\color{red}\left(\frac{d}{dv} \left(\ln\left(v\right)\right) \frac{d}{du} \left(2 u\right)\right)}$$자연로그 함수의 도함수는 $$$\frac{d}{dv} \left(\ln\left(v\right)\right) = \frac{1}{v}$$$:
$${\color{red}\left(\frac{d}{dv} \left(\ln\left(v\right)\right)\right)} \frac{d}{du} \left(2 u\right) = {\color{red}\left(\frac{1}{v}\right)} \frac{d}{du} \left(2 u\right)$$역치환:
$$\frac{\frac{d}{du} \left(2 u\right)}{{\color{red}\left(v\right)}} = \frac{\frac{d}{du} \left(2 u\right)}{{\color{red}\left(2 u\right)}}$$상수배 법칙 $$$\frac{d}{du} \left(c f{\left(u \right)}\right) = c \frac{d}{du} \left(f{\left(u \right)}\right)$$$을 $$$c = 2$$$와 $$$f{\left(u \right)} = u$$$에 적용합니다:
$$\frac{{\color{red}\left(\frac{d}{du} \left(2 u\right)\right)}}{2 u} = \frac{{\color{red}\left(2 \frac{d}{du} \left(u\right)\right)}}{2 u}$$멱법칙 $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$을 $$$n = 1$$$에 대해 적용하면, 즉 $$$\frac{d}{du} \left(u\right) = 1$$$:
$$\frac{{\color{red}\left(\frac{d}{du} \left(u\right)\right)}}{u} = \frac{{\color{red}\left(1\right)}}{u}$$따라서, $$$\frac{d}{du} \left(\ln\left(2 u\right)\right) = \frac{1}{u}$$$.
정답
$$$\frac{d}{du} \left(\ln\left(2 u\right)\right) = \frac{1}{u}$$$A