Can I take this product:
$$\frac{dL}{dt}\frac{d L}{d \dot{x}}$$
And factor out one of the $L$'s to get:
$$L\frac{d}{dt} \left( \frac{d L}{d \dot{x}}\right)$$
Where the operator $\frac{d}{dt}$ now operates on $\frac{d L}{d \dot{x}}$?
Is this allowed?
Thanks