Identity of dirac delta function

Question

Show that

1. if f is continuous

$\frac{d}{dx}(f(x)\delta(x)) = f(0)\delta'(x)$

2. if f is differentiable, use Leibnitz rule to conclude that

$\frac{d}{dx}(f(x)\delta(x)) = f'(x)\delta(x)+f(x)\delta'(x)$

I stuck at starting the proof. Should I use integration to do both proof?