1. ## Finding the Differential

Ian contributes \$2000 a month into an account earning interest at a rate of r/year compounded monthly. At the end of 25 years his account will be worth

$S=((24000[(1+(r/12))^(300)-1])/(r))$

Find the differential of S.

I'm having trouble calculating the derivative of this equation because there are so many mixed operations. I know the differential is S^(prime)(r)(dy/dx)x but how can I find the differential if I can't find the derivative?

2. Is the following the general formula for payment amount $P$, at annual interest rate $r$, with the interest compounded $m$ times a year over $t$ years, with $i\, =\, \frac{r}{m}$ and $n\, =\, tm$?

. . . . . $S\, =\, P\left[\frac{(1\, +\, i)^n\, -\, 1}{i}\right]$

If so, are you trying to find $dS$ in terms of $r$ and $dr$?

3. Yes, that's what I'm trying to do. I need to find out the change in S when r increases from 9% to some another percentage.

The question asks me to find how much more would Ian's account be worth if he earned 9.1% instead of 9%, and so on. I know that I can simply plug in .009 and .0091 into the equation and do S(.009)-S(.0091) but the question is asking me to estimate?

4. $

S\, =\, 24000\left[\frac{(1\, +\, i)^{300}\, -\, 1}{i}\right]
$

This is the equation the book gives me for the value of his account after 25 years and it wants me to find the derivative of it. When I plug it into my ti89 calculator my denominator gets astronomically large.