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 byy = k.a^{t}.e^{(bt)} for some a,b,k (using . for multiplication)

note that d/dt(a^{t}) = d/dt(e^{t.ln(a)}) = ln(a).e^{t.ln(a)} = a^{t}.ln(a)

and therefore by the chain rule

d/dt(e^{(bt)}) = b^{t}.e^{(bt)}.ln(b)

so using the product rule:

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

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