Yesterday we saw a graphical derivation of the centripetal acceleration in uniform circular motion. Today for those more mathematically oriented I would like to show an analytical derivation.
We simply follow the standard prescription we have defined
for obtaining the acceleration of a particle given the time
dependent position vector.
We start by writing the time dependent position vector
for the particle in uniform circular motion:
The corresponding velocity is
We notice that the magnitude of the velocity is
as derived previously and the direction of 
is indeed
tangential to the trajectory. We get the acceleration vector by taking
one more derivative:
And thus we end up with the same expression for the centripetal acceleration.