$$$x \sin{\left(x \right)}$$$의 이차 도함수
사용자 입력
$$$\frac{d^{2}}{dx^{2}} \left(x \sin{\left(x \right)}\right)$$$을(를) 구하시오.
풀이
제1도함수 $$$\frac{d}{dx} \left(x \sin{\left(x \right)}\right)$$$를 구하세요
$$$f{\left(x \right)} = x$$$와 $$$g{\left(x \right)} = \sin{\left(x \right)}$$$에 대해 곱의 미분법칙 $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$을 적용하십시오:
$${\color{red}\left(\frac{d}{dx} \left(x \sin{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x\right) \sin{\left(x \right)} + x \frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)}$$사인 함수의 도함수는 $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:
$$x {\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} + \sin{\left(x \right)} \frac{d}{dx} \left(x\right) = x {\color{red}\left(\cos{\left(x \right)}\right)} + \sin{\left(x \right)} \frac{d}{dx} \left(x\right)$$멱법칙 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$을 $$$n = 1$$$에 대해 적용하면, 즉 $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$x \cos{\left(x \right)} + \sin{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = x \cos{\left(x \right)} + \sin{\left(x \right)} {\color{red}\left(1\right)}$$따라서, $$$\frac{d}{dx} \left(x \sin{\left(x \right)}\right) = x \cos{\left(x \right)} + \sin{\left(x \right)}$$$.
다음으로, $$$\frac{d^{2}}{dx^{2}} \left(x \sin{\left(x \right)}\right) = \frac{d}{dx} \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)$$$
합/차의 도함수는 도함수들의 합/차이다:
$${\color{red}\left(\frac{d}{dx} \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x \cos{\left(x \right)}\right) + \frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)}$$사인 함수의 도함수는 $$$\frac{d}{dx} \left(\sin{\left(x \right)}\right) = \cos{\left(x \right)}$$$:
$${\color{red}\left(\frac{d}{dx} \left(\sin{\left(x \right)}\right)\right)} + \frac{d}{dx} \left(x \cos{\left(x \right)}\right) = {\color{red}\left(\cos{\left(x \right)}\right)} + \frac{d}{dx} \left(x \cos{\left(x \right)}\right)$$$$$f{\left(x \right)} = x$$$와 $$$g{\left(x \right)} = \cos{\left(x \right)}$$$에 대해 곱의 미분법칙 $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$을 적용하십시오:
$$\cos{\left(x \right)} + {\color{red}\left(\frac{d}{dx} \left(x \cos{\left(x \right)}\right)\right)} = \cos{\left(x \right)} + {\color{red}\left(\frac{d}{dx} \left(x\right) \cos{\left(x \right)} + x \frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)}$$멱법칙 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$을 $$$n = 1$$$에 대해 적용하면, 즉 $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$x \frac{d}{dx} \left(\cos{\left(x \right)}\right) + \cos{\left(x \right)} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} + \cos{\left(x \right)} = x \frac{d}{dx} \left(\cos{\left(x \right)}\right) + \cos{\left(x \right)} {\color{red}\left(1\right)} + \cos{\left(x \right)}$$코사인의 도함수는 $$$\frac{d}{dx} \left(\cos{\left(x \right)}\right) = - \sin{\left(x \right)}$$$입니다:
$$x {\color{red}\left(\frac{d}{dx} \left(\cos{\left(x \right)}\right)\right)} + 2 \cos{\left(x \right)} = x {\color{red}\left(- \sin{\left(x \right)}\right)} + 2 \cos{\left(x \right)}$$따라서, $$$\frac{d}{dx} \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) = - x \sin{\left(x \right)} + 2 \cos{\left(x \right)}$$$.
따라서 $$$\frac{d^{2}}{dx^{2}} \left(x \sin{\left(x \right)}\right) = - x \sin{\left(x \right)} + 2 \cos{\left(x \right)}$$$.
정답
$$$\frac{d^{2}}{dx^{2}} \left(x \sin{\left(x \right)}\right) = - x \sin{\left(x \right)} + 2 \cos{\left(x \right)}$$$A