Let A → Ar be the map that takes a positive definite matrix to its rth power, and let DAr be the Fréchet derivative of this map. It is known that ∥DAr∥ = ∥rAr−1∥ if r ≤ 1 or if r ≥ 2, and also that ...

In other words you add one to the power, divide by the new power and also by the derivative of what's inside the bracket. Don't forget to add the constant of integration.