Derivative of $$$x^{2} + \sqrt{2} x + 1$$$

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

$${\color{red}\left(\frac{d}{dx} \left(x^{2} + \sqrt{2} x + 1\right)\right)} = {\color{red}\left(\frac{d}{dx} \left(x^{2}\right) + \frac{d}{dx} \left(\sqrt{2} x\right) + \frac{d}{dx} \left(1\right)\right)}$$

Apply the constant multiple rule $$$\frac{d}{dx} \left(c f{\left(x \right)}\right) = c \frac{d}{dx} \left(f{\left(x \right)}\right)$$$ with $$$c = \sqrt{2}$$$ and $$$f{\left(x \right)} = x$$$:

$${\color{red}\left(\frac{d}{dx} \left(\sqrt{2} x\right)\right)} + \frac{d}{dx} \left(1\right) + \frac{d}{dx} \left(x^{2}\right) = {\color{red}\left(\sqrt{2} \frac{d}{dx} \left(x\right)\right)} + \frac{d}{dx} \left(1\right) + \frac{d}{dx} \left(x^{2}\right)$$

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

$$\sqrt{2} {\color{red}\left(\frac{d}{dx} \left(x\right)\right)} + \frac{d}{dx} \left(1\right) + \frac{d}{dx} \left(x^{2}\right) = \sqrt{2} {\color{red}\left(1\right)} + \frac{d}{dx} \left(1\right) + \frac{d}{dx} \left(x^{2}\right)$$

Apply the power rule $$$\frac{d}{dx} \left(x^{n}\right) = n x^{n - 1}$$$ with $$$n = 2$$$:

$${\color{red}\left(\frac{d}{dx} \left(x^{2}\right)\right)} + \frac{d}{dx} \left(1\right) + \sqrt{2} = {\color{red}\left(2 x\right)} + \frac{d}{dx} \left(1\right) + \sqrt{2}$$

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

$$2 x + {\color{red}\left(\frac{d}{dx} \left(1\right)\right)} + \sqrt{2} = 2 x + {\color{red}\left(0\right)} + \sqrt{2}$$

Thus, $$$\frac{d}{dx} \left(x^{2} + \sqrt{2} x + 1\right) = 2 x + \sqrt{2}$$$.