## Exponential Function Product Rule

### A first definition of exponential and logarithmic functions

Motivation from compound interest Increase the number of times that the interest is compounded each year ($n$), so as to increase the final amount of money ($A$)
$$A = P \left(1 + \frac{r}{n}\right)^{nt}.$$ $t$ and $r$ are the number of years and the interest rate per annum.
As $n$ becomes large, we can approximate the amount by
$$A = P e^{rt}.$$ The value of $A$ in the above formula is the amount compounded continuously.
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