# Thread: Finding the Differential

1. ## Finding the Differential

Ian contributes $2000 a month into an account earning interest at a rate of r/year compounded monthly. At the end of 25 years his account will be worth$\displaystyle S=((24000[(1+(r/12))^(300)-1])/(r))$Find the differential of S. I'm having trouble calculating the derivative of this equation because there are so many mixed operations. I know the differential is S^(prime)(r)(dy/dx)x but how can I find the differential if I can't find the derivative? 2. Is the following the general formula for payment amount$\displaystyle P$, at annual interest rate$\displaystyle r$, with the interest compounded$\displaystyle m$times a year over$\displaystyle t$years, with$\displaystyle i\, =\, \frac{r}{m}$and$\displaystyle n\, =\, tm$? . . . . .$\displaystyle S\, =\, P\left[\frac{(1\, +\, i)^n\, -\, 1}{i}\right]$If so, are you trying to find$\displaystyle dS$in terms of$\displaystyle r$and$\displaystyle dr$? 3. Yes, that's what I'm trying to do. I need to find out the change in S when r increases from 9% to some another percentage. The question asks me to find how much more would Ian's account be worth if he earned 9.1% instead of 9%, and so on. I know that I can simply plug in .009 and .0091 into the equation and do S(.009)-S(.0091) but the question is asking me to estimate? 4.$\displaystyle

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.