Originally Posted by

**Chris L T521** The second to last line isn't correct. We can't assume its linear! However, note that $\displaystyle \frac{\,d}{\,dt}\left[ap\right]+\frac{\,d}{\,dt}\left[bq\right]=a\frac{\,dp}{\,dt}+b\frac{\,dq}{\,dt}=a T\!\left(p\right)+bT\!\left(q\right)$.

Taking $\displaystyle T\!\left(ap+bq\right)$ takes care of the constant muliple part as well. So you only need to check this linear combination, instead of the sum and constant multiple parts separately (in other words, its easier and quicker to do both steps in one step).

Does this make sense?