What will be the differential function of √(x² + 2)?

The correct option is x/√(x² + 2) dx.

To find the differential of the function, we let:

$y=\sqrt{x^2+2}$

The differential of $y$ is given by the formula:
$dy=\frac{dy}{dx}dx$

First, we find the derivative of $y$ with respect to $x$ using the chain rule. Let $u=x^2+2$, so $y=\sqrt{u}=u^{1/2}$.

Applying the chain rule:
$\frac{dy}{dx}=\frac{dy}{du}\cdot \frac{du}{dx}$

We compute the individual derivatives:
$\frac{dy}{du}=\frac{1}{2}{u}^{-1/2}=\frac{1}{2\sqrt{u}}$
$\frac{du}{dx}=2x$

Substitute these back into the chain rule formula:
$\frac{dy}{dx}=\frac{1}{2\sqrt{x^2+2}}\cdot \left(2x\right)$

Simplifying the expression by canceling the factor of 2 in the numerator and denominator:
$\frac{dy}{dx}=\frac{x}{\sqrt{x^2+2}}$

Therefore, the differential of the function is:
$dy=\frac{x}{\sqrt{x^2+2}}dx$