  In unpublished notes Pila proposed a Modular Zilber-Pink with Derivatives
(MZPD) conjecture, which is a Zilber-Pink type statement for the modular
$j$-function and its derivatives. In this article we define D-special
varieties, then state and prove two functional (differential) analogues of the
MZPD conjecture for those varieties. In particular, we prove a weak version of
MZPD. As a special case of our results, we obtain a functional Modular
Andr\'e-Oort with Derivatives statement. The main tools used in the paper come
from (model theoretic) differential algebra and complex analytic geometry, and
the Ax-Schanuel theorem for the $j$-function and its derivatives (established
by Pila and Tsimerman) plays a crucial role in our proofs. In the proof of the
second Zilber-Pink type theorem we also use an Existential Closedness statement
for the differential equation of the $j$-function.
