|
|
|
|
|
|
|
|
|
|
|
|
$$$\cos{\left(x y \right)}$$$ 关于 $$$y$$$ 的导数
您的输入
求$$$\frac{d}{dy} \left(\cos{\left(x y \right)}\right)$$$。
解答
函数$$$\cos{\left(x y \right)}$$$是两个函数$$$f{\left(u \right)} = \cos{\left(u \right)}$$$和$$$g{\left(y \right)} = x y$$$的复合$$$f{\left(g{\left(y \right)} \right)}$$$。
应用链式法则 $$$\frac{d}{dy} \left(f{\left(g{\left(y \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dy} \left(g{\left(y \right)}\right)$$$：
$${\color{red}\left(\frac{d}{dy} \left(\cos{\left(x y \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right) \frac{d}{dy} \left(x y\right)\right)}$$余弦函数的导数是$$$\frac{d}{du} \left(\cos{\left(u \right)}\right) = - \sin{\left(u \right)}$$$：
$${\color{red}\left(\frac{d}{du} \left(\cos{\left(u \right)}\right)\right)} \frac{d}{dy} \left(x y\right) = {\color{red}\left(- \sin{\left(u \right)}\right)} \frac{d}{dy} \left(x y\right)$$返回到原变量：
$$- \sin{\left({\color{red}\left(u\right)} \right)} \frac{d}{dy} \left(x y\right) = - \sin{\left({\color{red}\left(x y\right)} \right)} \frac{d}{dy} \left(x y\right)$$对 $$$c = x$$$ 和 $$$f{\left(y \right)} = y$$$ 应用常数倍法则 $$$\frac{d}{dy} \left(c f{\left(y \right)}\right) = c \frac{d}{dy} \left(f{\left(y \right)}\right)$$$：
$$- \sin{\left(x y \right)} {\color{red}\left(\frac{d}{dy} \left(x y\right)\right)} = - \sin{\left(x y \right)} {\color{red}\left(x \frac{d}{dy} \left(y\right)\right)}$$应用幂法则 $$$\frac{d}{dy} \left(y^{n}\right) = n y^{n - 1}$$$，取 $$$n = 1$$$，也就是说，$$$\frac{d}{dy} \left(y\right) = 1$$$：
$$- x \sin{\left(x y \right)} {\color{red}\left(\frac{d}{dy} \left(y\right)\right)} = - x \sin{\left(x y \right)} {\color{red}\left(1\right)}$$因此，$$$\frac{d}{dy} \left(\cos{\left(x y \right)}\right) = - x \sin{\left(x y \right)}$$$。
答案
$$$\frac{d}{dy} \left(\cos{\left(x y \right)}\right) = - x \sin{\left(x y \right)}$$$A