On intersection of derivatives images for APN functions

The class of APN functions is considered in the paper. A vector Boolean function $F$ in $n$ variables from the set of all binary vectors of length $n$ to itself is called an APN function if the equation $F(x)\oplus F(x\oplus a)=b$ has at most $2$ solutions for any vectors $a,b$, where $a$ is a nonzero vector. A derivative of the function $F$ in the direction of $a$ is a Boolean function $D_aF(x)=F(x)\oplus F(x\oplus a)$. Two questions about intersections of the value sets for derivatives of two APN functions are proposed. The first one is about the minimal cardinality of such intersections. The second question is what a relationship these two APN functions have if the value sets of all directional derivatives of them pairwise coincide. Some partial results about both questions are obtained.