$$$\ln\left(6 x^{4}\right)$$$의 도함수
사용자 입력
$$$\frac{d}{dx} \left(\ln\left(6 x^{4}\right)\right)$$$을(를) 구하시오.
풀이
함수 $$$\ln\left(6 x^{4}\right)$$$는 두 함수 $$$f{\left(u \right)} = \ln\left(u\right)$$$와 $$$g{\left(x \right)} = 6 x^{4}$$$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:
$${\color{red}\left(\frac{d}{dx} \left(\ln\left(6 x^{4}\right)\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(6 x^{4}\right)\right)}$$자연로그 함수의 도함수는 $$$\frac{d}{du} \left(\ln\left(u\right)\right) = \frac{1}{u}$$$:
$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(6 x^{4}\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(6 x^{4}\right)$$역치환:
$$\frac{\frac{d}{dx} \left(6 x^{4}\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dx} \left(6 x^{4}\right)}{{\color{red}\left(6 x^{4}\right)}}$$상수배 법칙 $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$을 $$$c = 6$$$와 $$$f{\left(x \right)} = x^{4}$$$에 적용합니다:
$$\frac{{\color{red}\left(\frac{d}{dx} \left(6 x^{4}\right)\right)}}{6 x^{4}} = \frac{{\color{red}\left(6 \frac{d}{dx} \left(x^{4}\right)\right)}}{6 x^{4}}$$거듭제곱법칙 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$을 $$$n = 4$$$에 적용합니다:
$$\frac{{\color{red}\left(\frac{d}{dx} \left(x^{4}\right)\right)}}{x^{4}} = \frac{{\color{red}\left(4 x^{3}\right)}}{x^{4}}$$따라서, $$$\frac{d}{dx} \left(\ln\left(6 x^{4}\right)\right) = \frac{4}{x}$$$.
정답
$$$\frac{d}{dx} \left(\ln\left(6 x^{4}\right)\right) = \frac{4}{x}$$$A