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$$$\sin{\left(5 \theta \right)}$$$의 도함수
관련 계산기: 로그 미분 계산기, 암시적 미분 계산기 (단계별 풀이)
사용자 입력
$$$\frac{d}{d\theta} \left(\sin{\left(5 \theta \right)}\right)$$$을(를) 구하시오.
풀이
함수 $$$\sin{\left(5 \theta \right)}$$$는 두 함수 $$$f{\left(u \right)} = \sin{\left(u \right)}$$$와 $$$g{\left(\theta \right)} = 5 \theta$$$의 합성함수 $$$f{\left(g{\left(\theta \right)} \right)}$$$이다.
연쇄법칙 $$$\frac{d}{d\theta} \left(f{\left(g{\left(\theta \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{d\theta} \left(g{\left(\theta \right)}\right)$$$을(를) 적용하십시오:
$${\color{red}\left(\frac{d}{d\theta} \left(\sin{\left(5 \theta \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right) \frac{d}{d\theta} \left(5 \theta\right)\right)}$$사인 함수의 도함수는 $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:
$${\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right)\right)} \frac{d}{d\theta} \left(5 \theta\right) = {\color{red}\left(\cos{\left(u \right)}\right)} \frac{d}{d\theta} \left(5 \theta\right)$$역치환:
$$\cos{\left({\color{red}\left(u\right)} \right)} \frac{d}{d\theta} \left(5 \theta\right) = \cos{\left({\color{red}\left(5 \theta\right)} \right)} \frac{d}{d\theta} \left(5 \theta\right)$$상수배 법칙 $$$\frac{d}{d\theta} \left(c f{\left(\theta \right)}\right) = c \frac{d}{d\theta} \left(f{\left(\theta \right)}\right)$$$을 $$$c = 5$$$와 $$$f{\left(\theta \right)} = \theta$$$에 적용합니다:
$$\cos{\left(5 \theta \right)} {\color{red}\left(\frac{d}{d\theta} \left(5 \theta\right)\right)} = \cos{\left(5 \theta \right)} {\color{red}\left(5 \frac{d}{d\theta} \left(\theta\right)\right)}$$멱법칙 $$$\frac{d}{d\theta} \left(\theta^{n}\right) = n \theta^{n - 1}$$$을 $$$n = 1$$$에 대해 적용하면, 즉 $$$\frac{d}{d\theta} \left(\theta\right) = 1$$$:
$$5 \cos{\left(5 \theta \right)} {\color{red}\left(\frac{d}{d\theta} \left(\theta\right)\right)} = 5 \cos{\left(5 \theta \right)} {\color{red}\left(1\right)}$$따라서, $$$\frac{d}{d\theta} \left(\sin{\left(5 \theta \right)}\right) = 5 \cos{\left(5 \theta \right)}$$$.
정답
$$$\frac{d}{d\theta} \left(\sin{\left(5 \theta \right)}\right) = 5 \cos{\left(5 \theta \right)}$$$A