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$$$e^{x y z}$$$ 對 $$$x$$$ 的導數
您的輸入
求$$$\frac{d}{dx} \left(e^{x y z}\right)$$$。
解答
函數 $$$e^{x y z}$$$ 是兩個函數 $$$f{\left(u \right)} = e^{u}$$$ 與 $$$g{\left(x \right)} = x y z$$$ 之複合 $$$f{\left(g{\left(x \right)} \right)}$$$。
應用鏈式法則 $$$\frac{d}{dx} \left(f{\left(g{\left(x \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dx} \left(g{\left(x \right)}\right)$$$：
$${\color{red}\left(\frac{d}{dx} \left(e^{x y z}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(e^{u}\right) \frac{d}{dx} \left(x y z\right)\right)}$$指數函數的導數為 $$$\frac{d}{du} \left(e^{u}\right) = e^{u}$$$：
$${\color{red}\left(\frac{d}{du} \left(e^{u}\right)\right)} \frac{d}{dx} \left(x y z\right) = {\color{red}\left(e^{u}\right)} \frac{d}{dx} \left(x y z\right)$$返回原變數：
$$e^{{\color{red}\left(u\right)}} \frac{d}{dx} \left(x y z\right) = e^{{\color{red}\left(x y z\right)}} \frac{d}{dx} \left(x y z\right)$$套用常數倍法則 $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$，使用 $$$c = y z$$$ 與 $$$f{\left(x \right)} = x$$$：
$$e^{x y z} {\color{red}\left(\frac{d}{dx} \left(x y z\right)\right)} = e^{x y z} {\color{red}\left(y z \frac{d}{dx} \left(x\right)\right)}$$套用冪次法則 $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$，取 $$$n = 1$$$，也就是 $$$\frac{d}{dx} \left(x\right) = 1$$$：
$$y z e^{x y z} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} = y z e^{x y z} {\color{red}\left(1\right)}$$因此，$$$\frac{d}{dx} \left(e^{x y z}\right) = y z e^{x y z}$$$。
答案
$$$\frac{d}{dx} \left(e^{x y z}\right) = y z e^{x y z}$$$A