Help understanding Weyl's proof of Heisenberg's Inequality https://www.math.unl.edu/~scohn1/8423/heisenberg.pdf
Can someone help me understand how they go from line to line in this proof? I'm confused by it. The proof is on the last page. Specifically how do they go from $$\left(\int \left\vert xf^*(x)f'(x)\right\vert\, dx\right)^2 \geq \frac14 \left(\int x\frac{d}{dx}\left\vert f(x)\right\vert^2\,dx\right)^2 = \frac14\left(\int \left\vert f(x)\right\vert^2 \, dx\right)^2$$ Also why were they able make the following substitution? $$\int \left\vert\widehat{f'}(k)\right\vert^2\, dk = \int\left\vert f'(x)\right\vert^2\,dk$$

For the first question, note that $\frac{1}{2}\frac{d}{dx}|f|^2 = ff'$.  For the second question, use the Plancherel Theorem.
