Real analysis Folland problem 8.18

Question

Suppose $f\in L^2(\mathbb R)$.
a. The $L^2$ derivative $f'$ (in the sense of Exercises 8 and 9) exists iff $\xi \widehat f\in L^2$, in which case $\widehat{f'}(\xi)=2\pi i\xi \widehat{f}(\xi)$.
b. If the $L^2$ derivative $f'$ exists then $$\left[\int|f(x)|^2\,\mathrm{d}x\right]^2 \le 4 \int x|f(x)|^2 \,\mathrm{d}x \int |f'(x)|^2 \,\mathrm{d}x.$$ (If the integrals on the right are finite then one can integrate by parts to obtain $\int|f|^2=-2\operatorname{Re}x\overline ff'$.)
c. (Heisenberg inequality) For any $b,\beta\in\mathbb R$, $$\int (x-b)^2 |f(x)|^2 \,\mathrm{d}x \int (\xi-\beta)^2 |\widehat f(\xi)|^2 \,\mathrm{d}\xi \ge \frac{\|f\|^4_2}{16\pi^2}.$$

The definition of $L^{p}$ derivative:

Suppose that $f\in L^p(\mathbb R)$.If there exists $h\in L^p(\mathbb R)$ such that
$$\lim_{y\to 0}\Vert \frac{f(x+y)-f(x)}{y}-h(x)\Vert_{p}=0,$$ we call $h$ the (strong)$L^{p}$ derivative of $f$.

Actually I get stuck at the beginning of the question, since I just have no idea with how to use the condition of the $L^2$ derivative. I tried to find some useful theorems that might be useful on Folland's book(like The Plancherel Theorem), but they all need the condition that f $f\in L^1$.

(I have added the definition of $L^{p}$ derivative.)