What are the differentials for the given functions and rules, including product rule and basic functions?

The correct option is (B) $2xdx$.

2. If $V = x^3$, then differential of $V$ is:

The correct option is (A) $3x^2 dx$.

3. $d(xy) =$

Let $u = x$ and $v = y$. Then $du = dx$ and $dv = dy$.

The correct differential is $\boxed{x \, dy + y \, dx}$.

4. Differential of $\sqrt{y}$ is:

The correct differential is $\boxed{\frac{1}{2\sqrt{y}} \, dy}$.

5. Value of $dy$, for $y = x^2$ and $x$ char:
This question is incomplete as the full context for the change in $x$ is not provided.

6. Differential of $\sqrt{x}$ is:

The correct differential is $\boxed{\frac{1}{2\sqrt{x}} \, dx}$.

7. $\int \frac{\sin 2x}{4 \sin x} dx =$

The correct integral is $\boxed{\frac{1}{2} \sin x + C}$.

8. $\int \tan \frac{\pi}{4} dx$

The correct integral is $\boxed{x + C}$.