$$$\sin{\left(\frac{2 \ln\left(x\right)}{3} \right)}$$$의 도함수

함수 $$$\sin{\left(\frac{2 \ln\left(x\right)}{3} \right)}$$$는 두 함수 $$$f{\left(u \right)} = \sin{\left(u \right)}$$$와 $$$g{\left(x \right)} = \frac{2 \ln\left(x\right)}{3}$$$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:

$${\color{red}\left(\frac{d}{dx} \left(\sin{\left(\frac{2 \ln\left(x\right)}{3} \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right) \frac{d}{dx} \left(\frac{2 \ln\left(x\right)}{3}\right)\right)}$$

사인 함수의 도함수는 $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:

$${\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right)\right)} \frac{d}{dx} \left(\frac{2 \ln\left(x\right)}{3}\right) = {\color{red}\left(\cos{\left(u \right)}\right)} \frac{d}{dx} \left(\frac{2 \ln\left(x\right)}{3}\right)$$

역치환:

$$\cos{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(\frac{2 \ln\left(x\right)}{3}\right) = \cos{\left({\color{red}\left(\frac{2 \ln\left(x\right)}{3}\right)} \right)} \frac{d}{dx} \left(\frac{2 \ln\left(x\right)}{3}\right)$$

상수배 법칙 $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$을 $$$c = \frac{2}{3}$$$와 $$$f{\left(x \right)} = \ln\left(x\right)$$$에 적용합니다:

$$\cos{\left(\frac{2 \ln\left(x\right)}{3} \right)} {\color{red}\left(\frac{d}{dx} \left(\frac{2 \ln\left(x\right)}{3}\right)\right)} = \cos{\left(\frac{2 \ln\left(x\right)}{3} \right)} {\color{red}\left(\frac{2 \frac{d}{dx} \left(\ln\left(x\right)\right)}{3}\right)}$$

자연로그 함수의 도함수는 $$$\frac{d}{dx} \left(\ln\left(x\right)\right) = \frac{1}{x}$$$:

$$\frac{2 \cos{\left(\frac{2 \ln\left(x\right)}{3} \right)} {\color{red}\left(\frac{d}{dx} \left(\ln\left(x\right)\right)\right)}}{3} = \frac{2 \cos{\left(\frac{2 \ln\left(x\right)}{3} \right)} {\color{red}\left(\frac{1}{x}\right)}}{3}$$

따라서, $$$\frac{d}{dx} \left(\sin{\left(\frac{2 \ln\left(x\right)}{3} \right)}\right) = \frac{2 \cos{\left(\frac{2 \ln\left(x\right)}{3} \right)}}{3 x}$$$.