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