Spaghettification inside a spherically symmetric black hole

[PraisetheSun]

I need to find the vectors for time and radius that describe a space-like 4-acceleration of an observer falling radially into a spherically-symmetric black hole. Previous to this question, the values of the real time derivatives for time and radius were derived to be:
dt/dτ = (1-2m/r)^-1

and

dr/dτ = -(2m/r)^1/2

In order to form the vectors needed for 4 acceleration, is it possible to differentiate these equations with respect to τ(real time) again? I am stumped!

http://imgur.com/gallery/2i0PbnB/new

 

[PraisetheSun]

Here is the question in more detail. I am on part 2