What will be the differential function of log(x² + 4)?

The correct option is: 2x/(x² + 4) dx

To find the differential function of $\log (x^2+4)$, we can apply the chain rule of differentiation along with the definition of a differential.

Let the given function be:
$y=\log (x^2+4)$

The differential $dy$ of a function $y=f\left(x\right)$ is given by the formula:
$dy=\frac{dy}{dx}dx$

First, let's find the derivative $\frac{dy}{dx}$ using the chain rule. The chain rule states that if $y=\log \left(u\right)$ where $u=x^2+4$, then:
$\frac{dy}{dx}=\frac{d}{du}\left[\log \right(u\left)\right]\cdot \frac{du}{dx}$

Now, we evaluate the derivative of the outer logarithmic function:
$\frac{d}{du}\left[\log \right(u\left)\right]=\frac{1}{u}$

Next, we find the derivative of the inner function $u=x^2+4$ with respect to $x$:
$\frac{du}{dx}=\frac{d}{dx}(x^2+4)=2x$

Multiplying these two components together according to the chain rule gives us:
$\frac{dy}{dx}=\frac{1}{x^2+4}\cdot \left(2x\right)=\frac{2x}{x^2+4}$

Finally, we substitute this derivative back into the definition of the differential:
$dy=\frac{2x}{x^2+4}dx$