Derivative of $$$e^{x} + \sin{\left(y z \right)}$$$ with respect to $$$z$$$

The derivative of a sum/difference is the sum/difference of derivatives:

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

The function $$$\sin{\left(y z \right)}$$$ is the composition $$$f{\left(g{\left(z \right)} \right)}$$$ of two functions $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ and $$$g{\left(z \right)} = y z$$$.

Apply the chain rule $$$\frac{d}{dz} \left(f{\left(g{\left(z \right)} \right)}\right) = \frac{d}{du} \left(f{\left(u \right)}\right) \frac{d}{dz} \left(g{\left(z \right)}\right)$$$:

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

The derivative of the sine is $$$\frac{d}{du} \left(\sin{\left(u \right)}\right) = \cos{\left(u \right)}$$$:

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

Return to the old variable:

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

The derivative of a constant is $$$0$$$:

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

Apply the constant multiple rule $$$\frac{d}{dz} \left(c f{\left(z \right)}\right) = c \frac{d}{dz} \left(f{\left(z \right)}\right)$$$ with $$$c = y$$$ and $$$f{\left(z \right)} = z$$$:

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

Apply the power rule $$$\frac{d}{dz} \left(z^{n}\right) = n z^{n - 1}$$$ with $$$n = 1$$$, in other words, $$$\frac{d}{dz} \left(z\right) = 1$$$:

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

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