  In this paper, we focus on Strichartz's derivatives, a family of derivatives
including the normal derivative, on p.c.f. (post critically finite) fractals,
which are defined at vertex points in the graphs that approximate the fractal.
We obtain a weak continuity property of the derivatives for functions in the
domain of the Laplacian. For a function with zero normal derivative at any
fixed vertex, the derivatives, including the normal derivatives of the
neighboring vertices will decay to zero. The optimal rates of approximations
are described and several non-trivial examples are provided to illustrate that
our estimates are sharp. We also study the boundness property of derivatives
for functions in the domain of the Laplacian. A necessary condition for a
function having a weak tangent of order one at a vertex point is provided.
Furthermore, we give a counter-example of a conjecture of Strichartz on the
existence of higher order weak tangents.
