The formula for compound interest is:

As n increases, the number of compounding intervals increases. Is there a limit to the number of compounding periods possible per year?

Let . As n increases, the amount of time between compounding periods decreases, so as n →∞ , h → 0. Substitute h into the expression:

Examine :

Or in general:

with continuous _________ for r > 0 and continuous _________ for r < 0.

What is ?

1

ntr

A Pn

rh

n

xde

dx

1/

0lim 1

h

hh

x x u ud d due e e e

dx dx dx

Exponential and Logarithmic Functions

Ex 1: What is ?

Ex 2: Let a be a constant. What is ?

Recall y = ex and x = ey are inverses of each other. We can write this in logarithmic form:

What is the derivative of ln x? loga x?

xda

dx

23xde

dx

1Ex 3: ln

d

dx x

1 1 1ln ln log

lna

d d du dx u x

dx x dx u dx dx x a