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$$$2 x^{2} - 2^{\frac{2}{3}} x + \sqrt[3]{2}$$$의 도함수
관련 계산기: 로그 미분 계산기, 암시적 미분 계산기 (단계별 풀이)
사용자 입력
$$$\frac{d}{dx} \left(2 x^{2} - 2^{\frac{2}{3}} x + \sqrt[3]{2}\right)$$$을(를) 구하시오.
풀이
합/차의 도함수는 도함수들의 합/차이다:
$${\color{red}\left(\frac{d}{dx} \left(2 x^{2} - 2^{\frac{2}{3}} x + \sqrt[3]{2}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(2 x^{2}\right) - \frac{d}{dx} \left(2^{\frac{2}{3}} x\right) + \frac{d}{dx} \left(\sqrt[3]{2}\right)\right)}$$상수배 법칙 $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$을 $$$c = 2^{\frac{2}{3}}$$$와 $$$f{\left(x \right)} = x$$$에 적용합니다:
$$- {\color{red}\left(\frac{d}{dx} \left(2^{\frac{2}{3}} x\right)\right)} + \frac{d}{dx} \left(\sqrt[3]{2}\right) + \frac{d}{dx} \left(2 x^{2}\right) = - {\color{red}\left(2^{\frac{2}{3}} \frac{d}{dx} \left(x\right)\right)} + \frac{d}{dx} \left(\sqrt[3]{2}\right) + \frac{d}{dx} \left(2 x^{2}\right)$$멱법칙 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$을 $$$n = 1$$$에 대해 적용하면, 즉 $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$- 2^{\frac{2}{3}} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} + \frac{d}{dx} \left(\sqrt[3]{2}\right) + \frac{d}{dx} \left(2 x^{2}\right) = - 2^{\frac{2}{3}} {\color{red}\left(1\right)} + \frac{d}{dx} \left(\sqrt[3]{2}\right) + \frac{d}{dx} \left(2 x^{2}\right)$$상수배 법칙 $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$을 $$$c = 2$$$와 $$$f{\left(x \right)} = x^{2}$$$에 적용합니다:
$${\color{red}\left(\frac{d}{dx} \left(2 x^{2}\right)\right)} + \frac{d}{dx} \left(\sqrt[3]{2}\right) - 2^{\frac{2}{3}} = {\color{red}\left(2 \frac{d}{dx} \left(x^{2}\right)\right)} + \frac{d}{dx} \left(\sqrt[3]{2}\right) - 2^{\frac{2}{3}}$$거듭제곱법칙 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$을 $$$n = 2$$$에 적용합니다:
$$2 {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} + \frac{d}{dx} \left(\sqrt[3]{2}\right) - 2^{\frac{2}{3}} = 2 {\color{red}\left(2 x\right)} + \frac{d}{dx} \left(\sqrt[3]{2}\right) - 2^{\frac{2}{3}}$$상수의 도함수는 $$$0$$$입니다:
$$4 x + {\color{red}\left(\frac{d}{dx} \left(\sqrt[3]{2}\right)\right)} - 2^{\frac{2}{3}} = 4 x + {\color{red}\left(0\right)} - 2^{\frac{2}{3}}$$따라서, $$$\frac{d}{dx} \left(2 x^{2} - 2^{\frac{2}{3}} x + \sqrt[3]{2}\right) = 4 x - 2^{\frac{2}{3}}$$$.
정답
$$$\frac{d}{dx} \left(2 x^{2} - 2^{\frac{2}{3}} x + \sqrt[3]{2}\right) = 4 x - 2^{\frac{2}{3}}$$$A