I have just started learned some basic things about the binomial theorem, for fun.
I have seen that $1=\sum_{r=0}^{n} {n \choose r} x^r (1-x)^{n-r}$, and I would like to use this fact to prove that:
$nx =\sum_{r=0}^{n} r{n \choose r} x^r (1-x)^{n-r}$
and
$n(n-1)x^2 + nx =\sum_{r=0}^{n} r^2{n \choose r} x^r (1-x)^{n-r}$
What sort of techniques should I be using here? All I have learned in my reference so far is the binomial theorem, basic applications, examples, corollaries, and then that formula for $1$.
I attempted induction, substitution, differentiation, and haven't quite come up with the right answer yet.