How do you find rate of change for interest compounded quarterly

Question

10,000 initially in account, 4 percent interest compounded quarterly...

what is rate of change at 35 Years?

Answer 1

The formula is x (the amount)=10000(1+r)^t, where r is the quarterly rate and t the time in quarters.

r=4% per annum=4%/4=1% quarterly, so r=0.01. 1+r=1.01=e^ln(1.01).

x=10000(1.01)^t=10000e^tln(1.01),

dx/dt=10000ln(1.01)e^tln(1.01)=10000ln(1.01)(1.01)^t. This is the rate of change at time t quarters.

At 35 years, t=140 quarters. dx/dt=10000ln(1.01)(1.01)¹⁴⁰=400.71.

If the amount was dollars, then dx/dt is dollars/quarter. So in this case after 35 years, the rate of change is about $400 per quarter. In other words, the next quarter will see another $400 added to the amount.

After 35 years the original 10000 would have become 40271, so the rate of change is about 1%. ln(1.01) is about 0.01 or 1%.