I was wondering how those functions can be differentiated with respect to t. I found it a bit challenging. Can anyone found the derivative of those functions?
1. f(t) = exp(μt + (σ^2)(t^2)/2). ( exp is just e and the things in the bracket are the power of the e)
2. f(t) = (1 - βt)^α t< 1/β
3. f(t) = e^u((e^t) - 1)
y = (1 - bt)^a
Take logs to base e of both sides =>
ln(y) = a*ln(1 - bt)
Differentiate both side wrt t =>
(1/y)(dy/dt) = a*(-b/(1 - bt)) =>
dy/dt = a*(-b/(1 - bt))*(1 - bt)^a = -ab(1-bt)^(a - 1)
y = e^(ut + s^2t^2/2)
This is a function of a function - has form y = e^(f(t)) so differential is dy/dx = f'(t)*e^(f(t)) - so we have:
dy/dt = (u + s^2t)e^(ut + s^2t^2/2)
y = e^(u(e^t - 1)) - same thing again -
dy/dt = (u*e^t - 1)e^(u(e^t - 1))
Last edited by gnitsuk (2006-03-22 22:05:26)
it seems that differentiating those functions was like a piece of cake for you.
I think those functions can also be differentiated in some different way.
Anyways..thanks a lot..
I will post some integration questions and hope that you can evaluate those integrals