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In-Depth Information
The Exponential Function Is Its Own Derivative
d
dx
e
x
= e
x
.
It is this special property about the exponential function that makes it
unique and causes it to come up so frequently in applications. Anytime
the rate of change of some value is proportionate to the value itself, the
exponential function will almost certainly arise somewhere in the math
that describes the dynamics of the system.
The example most of us are familiar with is compound interest. Let P(t)
be the amount of money in your bank account at time t; assume the amount
is accruing interest. The rate of change per time interval—the amount of
interest earned—is proportional to the amount of money in your account.
The more money you have, the more interest you are earning, and the faster
it grows. Thus, the exponential function works its way into finance with
the equation P(t) = P
0
e
rt
, which describes the amount of money at any
given time t, assuming an initial amount P
0
grows at an interest rate of r,
where the interest is compounded continually.
You might have noticed that the Taylor series of e
x
is strikingly similar
to the series representation of sinx and cosx. This similarity hints at a
deep and surprising relationship between the exponential functions and the
trig functions, which we explore in Exercise 11.
We hope this brief encounter with Taylor series, although a bit out-
side of our main thrust, has sparked your interest in a mathematical tool
that is highly practical, in particular for its fundamental importance to all
sorts of approximation and numerical calculations in a computer. We also
hope it was an interesting non-trivial example of differentiation of a poly-
nomial. It also has given us a chance to discuss the derivatives of the sine,
cosine, and exponential functions; these derivatives come up again in later
sections.
11.4.7 The Chain Rule
The chain rule is the last rule of differentiation we discuss here. The chain
rule tells us how to determine the rate of change of a function when the
argument to that function is itself some other function we know how to
differentiate.
In the race between the tortoise and hare, we never really thought much
about exactly what our function x(t) measured, we just said it was the “po-
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