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$$$e^{x^{2}}$$$的导数
您的输入
求$$$\frac{d}{dx} \left(e^{x^{2}}\right)$$$。
解答
函数$$$e^{x^{2}}$$$是两个函数$$$f{\left(u \right)} = e^{u}$$$和$$$g{\left(x \right)} = x^{2}$$$的复合$$$f{\left(g{\left(x \right)} \right)}$$$。
应用链式法则 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$：
$${\color{red}\left(\frac{d}{dx} \left(e^{x^{2}}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(e^{u}\right) \frac{d}{dx} \left(x^{2}\right)\right)}$$指数函数的导数为 $$$\frac{d}{du} \left(e^{u}\right) = e^{u}$$$：
$${\color{red}\left(\frac{d}{du} \left(e^{u}\right)\right)} \frac{d}{dx} \left(x^{2}\right) = {\color{red}\left(e^{u}\right)} \frac{d}{dx} \left(x^{2}\right)$$返回到原变量：
$$e^{{\color{red}\left(u\right)}} \frac{d}{dx} \left(x^{2}\right) = e^{{\color{red}\left(x^{2}\right)}} \frac{d}{dx} \left(x^{2}\right)$$应用幂次法则 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$，其中 $$$n = 2$$$:
$$e^{x^{2}} {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} = e^{x^{2}} {\color{red}\left(2 x\right)}$$因此，$$$\frac{d}{dx} \left(e^{x^{2}}\right) = 2 x e^{x^{2}}$$$。
答案
$$$\frac{d}{dx} \left(e^{x^{2}}\right) = 2 x e^{x^{2}}$$$A
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