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