A) What exactly is the first one? I believe it is $\displaystyle \lfloor x\rfloor \sin^2(\pi x)$ with $\displaystyle \lfloor x\rfloor$ being the floor function.
B)$\displaystyle f:x\longmapsto\left\{\begin{array}{rcl} x^2\cos\left(\frac{\pi}{x}\right) & \mbox{if} & x\ne 0\\ 0 & \mbox{if} & x=0\end{array}\right.$
For the second one we see that the only problem is $\displaystyle f'(0)$, to ascertain its value/existence we revert back to the definitons.
$\displaystyle \begin{aligned}f'(0)&=\lim_{x\to 0}\frac{f(x)-f(0)}{x-0}\\
&=\lim_{x\to 0}\frac{x^2\cos\left(\frac{\pi}{x}\right)-0}{x}\\
&=\lim_{x\to 0}x\cos\left(\frac{\pi}{x}\right)\\
&=0\end{aligned}$
regarding A:
you descibed the function correctly
how do you choose what numbers to pick
after that i just put them into the formula
and if i get a final limit then its deferentiable on that point.
how do you choose what numbers to pick?
regarding b:
how did you know to check for 0
and only for 0
??