Problem understanding operator algebra

[Luna=Luna]

"It is left as a problem for the reader to show that if [S,T] commutes with S and T, then [e^{tT}, S] = -t[S,T]e^{tT}

I'm not sure if I'm missing something here, but i don't even see how it is possible to arrive at this answer.

I get:
[e^{tT}, S] = e^{tT}S - Se^{tT}

Then using the fact that [S,T] commutes with S and T this gives:

SST-STS = STS-TSS
and
TST-TTS = STT-TST

and see no way to go further.
One major thing is I don't even see how the factor of -t just appears in the identity?
[e^{tT}, S] = -t[S,T]e^{tT}

 

[MisterX]

The typical way to do something like this would be to use a Taylor expansion of the function of the operator, in this case e^{tT}. What you were trying to show is an example of a more general result.