$$$e^{x} + \sin{\left(y z \right)}$$$ 关于 $$$z$$$ 的导数

和/差的导数等于导数的和/差：

$${\color{red}\left(\frac{d}{dz} \left(e^{x} + \sin{\left(y z \right)}\right)\right)} = {\color{red}\left(\frac{d}{dz} \left(e^{x}\right) + \frac{d}{dz} \left(\sin{\left(y z \right)}\right)\right)}$$

常数的导数是$$$0$$$:

$${\color{red}\left(\frac{d}{dz} \left(e^{x}\right)\right)} + \frac{d}{dz} \left(\sin{\left(y z \right)}\right) = {\color{red}\left(0\right)} + \frac{d}{dz} \left(\sin{\left(y z \right)}\right)$$

函数$$$\sin{\left(y z \right)}$$$是两个函数$$$f{\left(u \right)} = \sin{\left(u \right)}$$$和$$$g{\left(z \right)} = y z$$$的复合$$$f{\left(g{\left(z \right)} \right)}$$$。

应用链式法则 $$$\frac{d}{dz} \left(f{\left(g{\left(z \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dz} \left(g{\left(z \right)}\right)$$$：

$${\color{red}\left(\frac{d}{dz} \left(\sin{\left(y z \right)}\right)\right)} = {\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right) \frac{d}{dz} \left(y z\right)\right)}$$

正弦函数的导数为 $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:

$${\color{red}\left(\frac{d}{du} \left(\sin{\left(u \right)}\right)\right)} \frac{d}{dz} \left(y z\right) = {\color{red}\left(\cos{\left(u \right)}\right)} \frac{d}{dz} \left(y z\right)$$

返回到原变量：

$$\cos{\left({\color{red}\left(u\right)} \right)} \frac{d}{dz} \left(y z\right) = \cos{\left({\color{red}\left(y z\right)} \right)} \frac{d}{dz} \left(y z\right)$$

对 $$$c = y$$$ 和 $$$f{\left(z \right)} = z$$$ 应用常数倍法则 $$$\frac{d}{dz} \left(c f{\left(z \right)}\right) = c \frac{d}{dz} \left(f{\left(z \right)}\right)$$$：

$$\cos{\left(y z \right)} {\color{red}\left(\frac{d}{dz} \left(y z\right)\right)} = \cos{\left(y z \right)} {\color{red}\left(y \frac{d}{dz} \left(z\right)\right)}$$

应用幂法则 $$$\frac{d}{dz} \left(z^{n}\right) = n z^{n - 1}$$$，取 $$$n = 1$$$，也就是说，$$$\frac{d}{dz} \left(z\right) = 1$$$：

$$y \cos{\left(y z \right)} {\color{red}\left(\frac{d}{dz} \left(z\right)\right)} = y \cos{\left(y z \right)} {\color{red}\left(1\right)}$$

因此，$$$\frac{d}{dz} \left(e^{x} + \sin{\left(y z \right)}\right) = y \cos{\left(y z \right)}$$$。