Adding Gravity component to Acceleration

first thing first, thank you so much for helping me with this problem.

I am working on a ROS-C++ Project. Problem:

i have a vector of positions (x, y, z) and a vector of time stamp with 10 times per second sampling(10Hz), from these two vector i need to output:

• A vector of velocities
• A vector of Accelerations
• A vector of Accelerations with gravity component

For the first two requests i did:

Velocity = dx/dt;

Acceleration = dv/dt

basically the medium Velocity and medium Acceleration.

Where: dx = Xi - Xi-1;

dt = ti - ti-1 and so on...

I really don't know if there are other or better solutions than these like if there is a C++ library that makes a function from a vector input and get the instantaneous velocity/acceleration but i am open to new solutions, maybe Alglib...

Now my real problem is considering the Gravity component in the last vector: i have accelerations but i don't think the solution would be cutting 9.8 from them :') I really searched everywhere but all i have are these vectors, i can also get the orientations values (x, y, z, w) but i don't think they are necessary for this problem?