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