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