Can the Lagrangian equation be simplified using the operator relation?

[Reshma]

Show that the Lagrange equations of the form
$${d\over dt}\left(\frac{\partial T}{\partial \dot{q_j}}\right) - \frac{\partial T}{\partial q_j} = Q_j$$
can also be written as
$$\frac{\partial \dot{T}}{\partial \dot{q_j}} - 2\frac{\partial T}{\partial q_j} = Q_j$$

Well it can be easily show that:
$${d\over dt}\left(\frac{\partial T}{\partial q_j \right)} = \frac{\partial \dot{T}}{\partial {q_j}}$$
Using the operator relation:
$$\frac{d}{dt} \frac{\partial}{\partial x} = \frac{\partial}{\partial x} \frac{d}{dt}$$

Can this be applied here?

 

[Reshma]

Come on folks, help me out on this. There's got to be a solution.