Tensor Calculus - How do I derive the formula for change of coordinates of a derivative?
The reason there are two terms is because of the product rule. We have $T_a = T_iJ^i_a$, so $\dfrac{\partial T_a}{\partial Z^b} = \dfrac{\partial}{\partial Z^b}(T_iJ^i_a) = \dfrac{\partial T_i}{\partial Z^b} J^i_a + T_i\dfrac{\partial J^i_a}{\partial Z^b} = \dfrac{\partial T_i}{\partial Z^j}J^i_aJ^j_b + T_iJ^i_{a,b}$.