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$$$e^{x^{2}}$$$の導関数
入力内容
$$$\frac{d}{dx} \left(e^{x^{2}}\right)$$$ を求めよ。
解答
関数$$$e^{x^{2}}$$$は、2つの関数$$$f{\left(u \right)} = e^{u}$$$と$$$g{\left(x \right)} = x^{2}$$$の合成$$$f{\left(g{\left(x \right)} \right)}$$$である。
連鎖律 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$ を適用する:
$${\color{red}\left(\frac{d}{dx} \left(e^{x^{2}}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(e^{u}\right) \frac{d}{dx} \left(x^{2}\right)\right)}$$指数関数の微分は$$$\frac{d}{du} \left(e^{u}\right) = e^{u}$$$です:
$${\color{red}\left(\frac{d}{du} \left(e^{u}\right)\right)} \frac{d}{dx} \left(x^{2}\right) = {\color{red}\left(e^{u}\right)} \frac{d}{dx} \left(x^{2}\right)$$元の変数に戻す:
$$e^{{\color{red}\left(u\right)}} \frac{d}{dx} \left(x^{2}\right) = e^{{\color{red}\left(x^{2}\right)}} \frac{d}{dx} \left(x^{2}\right)$$冪法則 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ を $$$n = 2$$$ に対して適用する:
$$e^{x^{2}} {\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} = e^{x^{2}} {\color{red}\left(2 x\right)}$$したがって、$$$\frac{d}{dx} \left(e^{x^{2}}\right) = 2 x e^{x^{2}}$$$。
解答
$$$\frac{d}{dx} \left(e^{x^{2}}\right) = 2 x e^{x^{2}}$$$A
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