Consider a function $\displaystyle \phi$ smooth, such that \int_{\mathbb R}\phi^2(t)dt=1, and put $\displaystyle \phi_n(t)=\sqrt n\phi(nt)$. Then $\displaystyle \lVert \phi_n\rVert_{L^2}=1$, but $\displaystyle \frac d{dt}\phi_n(t)=n^{\frac 32}\phi(nt)$ has a norm $\displaystyle L^2$ which cannot be bounded by a constant which doesn't depend on $\displaystyle n$.
For the second problem, note that $\displaystyle ||f||^2\geq ||\frac d{dt}f||^2_{L^2}$.