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$$$x e^{x}$$$의 도함수
사용자 입력
$$$\frac{d}{dx} \left(x e^{x}\right)$$$을(를) 구하시오.
풀이
$$$f{\left(x \right)} = x$$$와 $$$g{\left(x \right)} = e^{x}$$$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:
$${\color{red}\left(\frac{d}{dx} \left(x e^{x}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x\right) e^{x} + x \frac{d}{dx} \left(e^{x}\right)\right)}$$멱법칙 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$을 $$$n = 1$$$에 대해 적용하면, 즉 $$$\frac{d}{dx} \left(x\right) = 1$$$:
$$x \frac{d}{dx} \left(e^{x}\right) + e^{x} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = x \frac{d}{dx} \left(e^{x}\right) + e^{x} {\color{red}\left(1\right)}$$지수함수의 도함수는 $$$\frac{d}{dx} \left(e^{x}\right) = e^{x}$$$:
$$x {\color{red}\left(\frac{d}{dx} \left(e^{x}\right)\right)} + e^{x} = x {\color{red}\left(e^{x}\right)} + e^{x}$$간단히 하시오:
$$x e^{x} + e^{x} = \left(x + 1\right) e^{x}$$따라서, $$$\frac{d}{dx} \left(x e^{x}\right) = \left(x + 1\right) e^{x}$$$.
정답
$$$\frac{d}{dx} \left(x e^{x}\right) = \left(x + 1\right) e^{x}$$$A
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