Actually, there is an even stronger result, often called the interpolation theorem theorem, which follows from a well known-known theorem of Mittag-Leffler :
Let (z_n)$(z_n)$ be a sequence of complex numbers with no accumulationlimit point. For For each n$n$, let l(n)$l(n)$ be any integer greater or equal to 1 and (a_nk)$1$ and for (0 <= k <= l(n)$0 \leq k \leq l(n)$, let ) complex$(a_{n, k})$ be complex numbers. Then Then there exists an entire function g(z)$g$ such that
g^(k)(z_n) = a_nk$$g^{(k)}(z_n) = a_{n,k}$$
for every n>=1$n \geq 1$ and every 0 <= k <= l(n)$0 \leq k \leq l(n)$.
That is, you can fix values for the derivative at the z_j's$z_j$'s.