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$$$x e^{x}$$$ 的導數
您的輸入
求$$$\frac{d}{dx} \left(x e^{x}\right)$$$。
解答
將乘積法則 $$$\frac{d}{dx} \left(f{\left(x \right)} g{\left(x \right)}\right) = \frac{d}{dx} \left(f{\left(x \right)}\right) g{\left(x \right)} + f{\left(x \right)} \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ 應用於 $$$f{\left(x \right)} = x$$$ 和 $$$g{\left(x \right)} = e^{x}$$$：
$${\color{red}\left(\frac{d}{dx} \left(x e^{x}\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x\right) e^{x} + x \frac{d}{dx} \left(e^{x}\right)\right)}$$套用冪次法則 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$，取 $$$n = 1$$$，也就是 $$$\frac{d}{dx} \left(x\right) = 1$$$：
$$x \frac{d}{dx} \left(e^{x}\right) + e^{x} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = x \frac{d}{dx} \left(e^{x}\right) + e^{x} {\color{red}\left(1\right)}$$指數函數的導數為 $$$\frac{d}{dx} \left(e^{x}\right) = e^{x}$$$：
$$x {\color{red}\left(\frac{d}{dx} \left(e^{x}\right)\right)} + e^{x} = x {\color{red}\left(e^{x}\right)} + e^{x}$$化簡：
$$x e^{x} + e^{x} = \left(x + 1\right) e^{x}$$因此，$$$\frac{d}{dx} \left(x e^{x}\right) = \left(x + 1\right) e^{x}$$$。
答案
$$$\frac{d}{dx} \left(x e^{x}\right) = \left(x + 1\right) e^{x}$$$A
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