I think your notation is getting in your way - if you clean it up the problem will become a lot easier.
I will start by replacing the numbers with constants. What you have written (if I understand it right) can be represented by

y = k.at.e(bt) for some a,b,k (using . for multiplication)

note that d/dt(at) = d/dt(et.ln(a)) = ln(a).et.ln(a) = at.ln(a)
and therefore by the chain rule
d/dt(e(bt)) = bt.e(bt).ln(b)

so using the product rule:

d/dt(y) = k.{ at.ln(a).e(bt) + at.bt.ln(b).e(bt) }

now you need to evaluate a,b,k from your original expression and plug them in.