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$$$e^{\frac{u}{2}}$$$ 的導數
您的輸入
求$$$\frac{d}{du} \left(e^{\frac{u}{2}}\right)$$$。
解答
函數 $$$e^{\frac{u}{2}}$$$ 是兩個函數 $$$f{\left(v \right)} = e^{v}$$$ 與 $$$g{\left(u \right)} = \frac{u}{2}$$$ 之複合 $$$f{\left(g{\left(u \right)} \right)}$$$。
應用鏈式法則 $$$\frac{d}{du} \left(f{\left(g{\left(u \right)} \right)}\right) = \frac{d}{dv} \left(f{\left(v \right)}\right) \frac{d}{du} \left(g{\left(u \right)}\right)$$$：
$${\color{red}\left(\frac{d}{du} \left(e^{\frac{u}{2}}\right)\right)} = {\color{red}\left(\frac{d}{dv} \left(e^{v}\right) \frac{d}{du} \left(\frac{u}{2}\right)\right)}$$指數函數的導數為 $$$\frac{d}{dv} \left(e^{v}\right) = e^{v}$$$：
$${\color{red}\left(\frac{d}{dv} \left(e^{v}\right)\right)} \frac{d}{du} \left(\frac{u}{2}\right) = {\color{red}\left(e^{v}\right)} \frac{d}{du} \left(\frac{u}{2}\right)$$返回原變數：
$$e^{{\color{red}\left(v\right)}} \frac{d}{du} \left(\frac{u}{2}\right) = e^{{\color{red}\left(\frac{u}{2}\right)}} \frac{d}{du} \left(\frac{u}{2}\right)$$套用常數倍法則 $$$\frac{d}{du} \left(c f{\left(u \right)}\right) = c \frac{d}{du} \left(f{\left(u \right)}\right)$$$，使用 $$$c = \frac{1}{2}$$$ 與 $$$f{\left(u \right)} = u$$$：
$$e^{\frac{u}{2}} {\color{red}\left(\frac{d}{du} \left(\frac{u}{2}\right)\right)} = e^{\frac{u}{2}} {\color{red}\left(\frac{\frac{d}{du} \left(u\right)}{2}\right)}$$套用冪次法則 $$$\frac{d}{du} \left(u^{n}\right) = n u^{n - 1}$$$，取 $$$n = 1$$$，也就是 $$$\frac{d}{du} \left(u\right) = 1$$$：
$$\frac{e^{\frac{u}{2}} {\color{red}\left(\frac{d}{du} \left(u\right)\right)}}{2} = \frac{e^{\frac{u}{2}} {\color{red}\left(1\right)}}{2}$$因此，$$$\frac{d}{du} \left(e^{\frac{u}{2}}\right) = \frac{e^{\frac{u}{2}}}{2}$$$。
答案
$$$\frac{d}{du} \left(e^{\frac{u}{2}}\right) = \frac{e^{\frac{u}{2}}}{2}$$$A