$$$2^{n}$$$의 이차 도함수
사용자 입력
$$$\frac{d^{2}}{dn^{2}} \left(2^{n}\right)$$$을(를) 구하시오.
풀이
제1도함수 $$$\frac{d}{dn} \left(2^{n}\right)$$$를 구하세요
$$$m = 2$$$을 사용하여 지수법칙 $$$\frac{d}{dn} \left(m^{n}\right) = m^{n} \ln\left(m\right)$$$을 적용하십시오:
$${\color{red}\left(\frac{d}{dn} \left(2^{n}\right)\right)} = {\color{red}\left(2^{n} \ln\left(2\right)\right)}$$따라서, $$$\frac{d}{dn} \left(2^{n}\right) = 2^{n} \ln\left(2\right)$$$.
다음으로, $$$\frac{d^{2}}{dn^{2}} \left(2^{n}\right) = \frac{d}{dn} \left(2^{n} \ln\left(2\right)\right)$$$
상수배 법칙 $$$\frac{d}{dn} \left(c f{\left(n \right)}\right) = c \frac{d}{dn} \left(f{\left(n \right)}\right)$$$을 $$$c = \ln\left(2\right)$$$와 $$$f{\left(n \right)} = 2^{n}$$$에 적용합니다:
$${\color{red}\left(\frac{d}{dn} \left(2^{n} \ln\left(2\right)\right)\right)} = {\color{red}\left(\ln\left(2\right) \frac{d}{dn} \left(2^{n}\right)\right)}$$$$$m = 2$$$을 사용하여 지수법칙 $$$\frac{d}{dn} \left(m^{n}\right) = m^{n} \ln\left(m\right)$$$을 적용하십시오:
$$\ln\left(2\right) {\color{red}\left(\frac{d}{dn} \left(2^{n}\right)\right)} = \ln\left(2\right) {\color{red}\left(2^{n} \ln\left(2\right)\right)}$$따라서, $$$\frac{d}{dn} \left(2^{n} \ln\left(2\right)\right) = 2^{n} \ln^{2}\left(2\right)$$$.
따라서 $$$\frac{d^{2}}{dn^{2}} \left(2^{n}\right) = 2^{n} \ln^{2}\left(2\right)$$$.
정답
$$$\frac{d^{2}}{dn^{2}} \left(2^{n}\right) = 2^{n} \ln^{2}\left(2\right)$$$A
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