# Differentiation and Integration

• Mar 23rd 2006, 03:30 AM
Maitham
Differentiation and Integration
IS it possible to get an answer?

• Mar 23rd 2006, 05:37 AM
CaptainBlack
Quote:

Originally Posted by Maitham
IS it possible to get an answer?

Lets do the first derivative problem:

Differentiate:

$\displaystyle f(t)=\exp(\mu t + \sigma^2t^2/2)$

Here we need the chain rule:

$\displaystyle \frac{d}{dt}g(h(t))=\frac{dh}{dt}(t) \times \frac{dg}{dt}(h(t))$,

so put $\displaystyle h(t)=\mu t + \sigma^2t^2/2$ and $\displaystyle g(t)=exp(t)$.

Then:

$\displaystyle \frac{d}{dt}f(t)=[\mu+\sigma^2 t][\exp(\mu t + \sigma^2t^2/2)]$

RonL
• Mar 23rd 2006, 05:43 AM
CaptainBlack
Now lets do the first of the integrals:

$\displaystyle \int_0^{\infty} \frac{1}{\beta}e^{-x/\beta}dx$

The integrand is the derivative of $\displaystyle -e^{-x/\beta}$ so we know
(probably from the fundamentaly theorem of calculus) that:

$\displaystyle \int_0^{\infty} \frac{1}{\beta}e^{-x/\beta}dx=\left[-e^{-x/\beta}\right]_0^{\infty}=-0+1=1$

RonL
• Mar 27th 2006, 09:51 PM
Maitham
Thanks For the answer and i`ll try to do the rest and post the answer to check them for me if they are right, thats if it possible with you.