It is helpful to remember the definition of the exponential function in terms of a power series:
If we let x be imaginary,
(where 
is real),
then this can be written
which ought to remind you (doesn't it?) of the series expansions for the sinusoidal functions:
Note that these expansions perform a sort of ``leapfrog'' between even and odd terms. Putting them all together we can easily see that
and, since 
while
,
These simple equivalences are, to my mind, among the most
astonishing relationships in all of mathematics.
Why? Because they show an intimate relationship
between two functions which would seem at first glance
to have absolutely nothing in common: the monotonically
increasing or decreasing exponential function
and the sinusoidally oscillating
and
functions!
We can also invert the relationship and obtain a definition
for the 
and
functions in terms of exponentials:
Let's review the most important (for Physics, anyway)
property of the exponential function: it is its own derivative!
.
If x = kt then we ``pull out an extra factor of k''
with each derivative with respect to t:
This latter property (which, by the way, works just as well
for complex k as for real k)
establishes the connection between
the derivatives of
(for instance) and those of
and
:
You will recall how useful these second-derivative properties were in simple harmonic motion and waves. Well, you ain't seen nothing yet!