Let $g$ be a smooth function, i.e. a function which has derivatives of all orders. Recall that $g^{(k)}$ denotes the $k$-th derivative of $g$. For non-negative integers $n,k$, define the function $T_{n,k}(x)$ as follows: \[T_{n,k}(x)=\sum_{j=0}^n {n\choose j} (-1)^{n-j}g^{n-j}(x) (g^j)^{(k)}(x)\] (we set $g^0=1$ and $g^{(0)}=g$). For example, $T_{3,3}(x)=3g^2(x)g^{(3)}(x)-3g(x)(g^2)^{(3)}(x)+(g^3)^{(3)}(x)$.

a) Prove that $T_{n,k}(x)=0$ for any $n>k$ and any $x$.

b) Find a simple explicit formula for $T_{n,n}(x)$ (for example, $T_{3,3}(x)=6(g'(x))^3$. Check this!).