$$$\cot{\left(\frac{x}{2} \right)}$$$의 도함수

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

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

코탄젠트의 도함수는 $$$\frac{d}{du} \left(\cot{\left(u \right)}\right) = - \csc^{2}{\left(u \right)}$$$:

$${\color{red}\left(\frac{d}{du} \left(\cot{\left(u \right)}\right)\right)} \frac{d}{dx} \left(\frac{x}{2}\right) = {\color{red}\left(- \csc^{2}{\left(u \right)}\right)} \frac{d}{dx} \left(\frac{x}{2}\right)$$

역치환:

$$- \csc^{2}{\left({\color{red}\left(u\right)} \right)} \frac{d}{dx} \left(\frac{x}{2}\right) = - \csc^{2}{\left({\color{red}\left(\frac{x}{2}\right)} \right)} \frac{d}{dx} \left(\frac{x}{2}\right)$$

상수배 법칙 $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$을 $$$c = \frac{1}{2}$$$와 $$$f{\left(x \right)} = x$$$에 적용합니다:

$$- \csc^{2}{\left(\frac{x}{2} \right)} {\color{red}\left(\frac{d}{dx} \left(\frac{x}{2}\right)\right)} = - \csc^{2}{\left(\frac{x}{2} \right)} {\color{red}\left(\frac{\frac{d}{dx} \left(x\right)}{2}\right)}$$

멱법칙 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$을 $$$n = 1$$$에 대해 적용하면, 즉 $$$\frac{d}{dx} \left(x\right) = 1$$$:

$$- \frac{\csc^{2}{\left(\frac{x}{2} \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)}}{2} = - \frac{\csc^{2}{\left(\frac{x}{2} \right)} {\color{red}\left(1\right)}}{2}$$

간단히 하시오:

$$- \frac{\csc^{2}{\left(\frac{x}{2} \right)}}{2} = \frac{1}{\cos{\left(x \right)} - 1}$$

따라서, $$$\frac{d}{dx} \left(\cot{\left(\frac{x}{2} \right)}\right) = \frac{1}{\cos{\left(x \right)} - 1}$$$.