Derivative of $$$x + 1$$$

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

$${\color{red}\left(\frac{d}{dx} \left(x + 1\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x\right) + \frac{d}{dx} \left(1\right)\right)}$$

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

$${\color{red}\left(\frac{d}{dx} \left(x\right)\right)} + \frac{d}{dx} \left(1\right) = {\color{red}\left(1\right)} + \frac{d}{dx} \left(1\right)$$

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

$${\color{red}\left(\frac{d}{dx} \left(1\right)\right)} + 1 = {\color{red}\left(0\right)} + 1$$

Thus, $$$\frac{d}{dx} \left(x + 1\right) = 1$$$.