Product rule for a distribution and a function

Question

This should be simple enough; say f is a distribution and g is a function, show $(fg)'[t] = fg'[t] + f'g[t]$. I kept getting a negative sign when I was doing it myself, and looking up a solution, I can't justify the first step, the rest is fine. This is what I don't understand: $(fg)'[t] = (fg)[-t']$ How does that follow from the definitions of distributions? I don't get that one step. When I tried it first, I used the relation $fg[t] = f[gt]$ but that obviously doesn't hold when taking the derivative of fg. $(fg)'[t] = f'[gt] = -f[g't] - f[gt']= f'g[t] - fg'[t]$ which is wrong. What am I missing?