$$$x^{4} \cos{\left(x \right)}$$$의 도함수

$$$H{\left(x \right)} = x^{4} \cos{\left(x \right)}$$$라고 하자.

양변에 로그를 취합니다: $$$\ln\left(H{\left(x \right)}\right) = \ln\left(x^{4} \cos{\left(x \right)}\right)$$$.

로그의 성질을 이용하여 우변을 다시 쓰십시오: $$$\ln\left(H{\left(x \right)}\right) = 4 \ln\left(x\right) + \ln\left(\cos{\left(x \right)}\right)$$$

방정식의 양변을 각각 미분하시오: $$$\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right) = \frac{d}{dx} \left(4 \ln\left(x\right) + \ln\left(\cos{\left(x \right)}\right)\right)$$$

방정식의 좌변을 미분하세요.

함수 $$$\ln\left(H{\left(x \right)}\right)$$$는 두 함수 $$$f{\left(u \right)} = \ln\left(u\right)$$$와 $$$g{\left(x \right)} = H{\left(x \right)}$$$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:

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

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

$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(H{\left(x \right)}\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(H{\left(x \right)}\right)$$

역치환:

$$\frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{{\color{red}\left(u\right)}} = \frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{{\color{red}\left(H{\left(x \right)}\right)}}$$

따라서, $$$\frac{d}{dx} \left(\ln\left(H{\left(x \right)}\right)\right) = \frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{H{\left(x \right)}}$$$.

방정식의 우변을 미분하시오.

합/차의 도함수는 도함수들의 합/차이다:

$${\color{red}\left(\frac{d}{dx} \left(4 \ln\left(x\right) + \ln\left(\cos{\left(x \right)}\right)\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(4 \ln\left(x\right)\right) + \frac{d}{dx} \left(\ln\left(\cos{\left(x \right)}\right)\right)\right)}$$

함수 $$$\ln\left(\cos{\left(x \right)}\right)$$$는 두 함수 $$$f{\left(u \right)} = \ln\left(u\right)$$$와 $$$g{\left(x \right)} = \cos{\left(x \right)}$$$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:

$${\color{red}\left(\frac{d}{dx} \left(\ln\left(\cos{\left(x \right)}\right)\right)\right)} + \frac{d}{dx} \left(4 \ln\left(x\right)\right) = {\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right) \frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)} + \frac{d}{dx} \left(4 \ln\left(x\right)\right)$$

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

$${\color{red}\left(\frac{d}{du} \left(\ln\left(u\right)\right)\right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right) + \frac{d}{dx} \left(4 \ln\left(x\right)\right) = {\color{red}\left(\frac{1}{u}\right)} \frac{d}{dx} \left(\cos{\left(x \right)}\right) + \frac{d}{dx} \left(4 \ln\left(x\right)\right)$$

역치환:

$$\frac{d}{dx} \left(4 \ln\left(x\right)\right) + \frac{\frac{d}{dx} \left(\cos{\left(x \right)}\right)}{{\color{red}\left(u\right)}} = \frac{d}{dx} \left(4 \ln\left(x\right)\right) + \frac{\frac{d}{dx} \left(\cos{\left(x \right)}\right)}{{\color{red}\left(\cos{\left(x \right)}\right)}}$$

코사인의 도함수는 $$$\frac{d}{dx} \left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}$$$입니다:

$$\frac{d}{dx} \left(4 \ln\left(x\right)\right) + \frac{{\color{red}\left(\frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)}}{\cos{\left(x \right)}} = \frac{d}{dx} \left(4 \ln\left(x\right)\right) + \frac{{\color{red}\left(- \sin{\left(x \right)}\right)}}{\cos{\left(x \right)}}$$

상수배 법칙 $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$을 $$$c = 4$$$와 $$$f{\left(x \right)} = \ln\left(x\right)$$$에 적용합니다:

$$- \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + {\color{red}\left(\frac{d}{dx} \left(4 \ln\left(x\right)\right)\right)} = - \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + {\color{red}\left(4 \frac{d}{dx} \left(\ln\left(x\right)\right)\right)}$$

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

$$- \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + 4 {\color{red}\left(\frac{d}{dx} \left(\ln\left(x\right)\right)\right)} = - \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + 4 {\color{red}\left(\frac{1}{x}\right)}$$

간단히 하시오:

$$- \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + \frac{4}{x} = - \tan{\left(x \right)} + \frac{4}{x}$$

따라서, $$$\frac{d}{dx} \left(4 \ln\left(x\right) + \ln\left(\cos{\left(x \right)}\right)\right) = - \tan{\left(x \right)} + \frac{4}{x}$$$.

따라서 $$$\frac{\frac{d}{dx} \left(H{\left(x \right)}\right)}{H{\left(x \right)}} = - \tan{\left(x \right)} + \frac{4}{x}$$$.

따라서 $$$\frac{d}{dx} \left(H{\left(x \right)}\right) = \left(- \tan{\left(x \right)} + \frac{4}{x}\right) H{\left(x \right)} = x^{3} \left(- x \tan{\left(x \right)} + 4\right) \cos{\left(x \right)}$$$.