quantum mechanics - Factor 2 in Heisenberg Uncertainty Principle: Which formula is correct?

Some websites and textbooks refer to $$\Delta x \Delta p \geq \frac{\hbar}{2}$$ as the correct formula for the uncertainty principle whereas other sources use the formula $$\Delta x \Delta p \geq \hbar.$$ Question: Which one is correct and why?

The latter is used in the textbook "Physics II for Dummies" (German edition) for several examples and the author also derives that formula so I assume that this is not a typing error.

This is the mentioned derivation:

$\sin \theta = \frac{\lambda}{\Delta y}$

assuming $\theta$ is small:

$\tan \theta = \frac{\lambda}{\Delta y}$

de Broglie equation:

$\lambda = \frac{h}{p_x}$

$\Rightarrow \tan \theta \approx \frac{h}{p_x \cdot \Delta y}$

but also:

$\tan \theta = \frac{\Delta p_y}{p_x}$

equalize $\tan \theta$:

$\frac{h}{p_x \cdot \Delta y} \approx \frac{\Delta p_y}{p_x}$

$\Rightarrow \frac{h}{\Delta y} \approx \Delta p_y \Rightarrow \Delta p_y \Delta y \approx h$

$\Rightarrow \Delta p_y \Delta y \geq \frac{h}{2 \pi}$

$\Rightarrow \Delta p \Delta x \geq \frac{h}{2 \pi}$

So: Which one is correct and why?