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$$$\ln^{3}\left(x\right)$$$의 도함수
사용자 입력
$$$\frac{d}{dx} \left(\ln^{3}\left(x\right)\right)$$$을(를) 구하시오.
풀이
함수 $$$\ln^{3}\left(x\right)$$$는 두 함수 $$$f{\left(u \right)} = u^{3}$$$와 $$$g{\left(x \right)} = \ln\left(x\right)$$$의 합성함수 $$$f{\left(g{\left(x \right)} \right)}$$$이다.
연쇄법칙 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$을(를) 적용하십시오:
$${\color{red}\left(\frac{d}{dx} \left(\ln^{3}\left(x\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(u^{3}\right) \frac{d}{dx} \left(\ln\left(x\right)\right)\right)}$$거듭제곱법칙 $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$을 $$$n = 3$$$에 적용합니다:
$${\color{red}\left(\frac{d}{du} \left(u^{3}\right)\right)} \frac{d}{dx} \left(\ln\left(x\right)\right) = {\color{red}\left(3 u^{2}\right)} \frac{d}{dx} \left(\ln\left(x\right)\right)$$역치환:
$$3 {\color{red}\left(u\right)}^{2} \frac{d}{dx} \left(\ln\left(x\right)\right) = 3 {\color{red}\left(\ln\left(x\right)\right)}^{2} \frac{d}{dx} \left(\ln\left(x\right)\right)$$자연로그 함수의 도함수는 $$$\frac{d}{dx} \left(\ln\left(x\right)\right) = \frac{1}{x}$$$:
$$3 \ln^{2}\left(x\right) {\color{red}\left(\frac{d}{dx} \left(\ln\left(x\right)\right)\right)} = 3 \ln^{2}\left(x\right) {\color{red}\left(\frac{1}{x}\right)}$$따라서, $$$\frac{d}{dx} \left(\ln^{3}\left(x\right)\right) = \frac{3 \ln^{2}\left(x\right)}{x}$$$.
정답
$$$\frac{d}{dx} \left(\ln^{3}\left(x\right)\right) = \frac{3 \ln^{2}\left(x\right)}{x}$$$A