CALCULUS
Higher Concepts of Mathematics
f(x) = aebx
(5-9)
d [f(x)]
dx
abebx
These general techniques for finding the derivatives of functions are important for those who
perform detailed mathematical calculations for dynamic systems. For example, the designers of
nuclear facility systems need an understanding of these techniques, because these techniques are
not encountered in the day-to-day operation of a nuclear facility. As a result, the operators of
these facilities should understand what derivatives are in terms of a rate of change and a slope
of a graph, but they will not normally be required to find the derivatives of functions.
The notation d[f(x)]/dx is the common way to indicate the derivative of a function. In some
applications, the notation
is used. In other applications, the so-called dot notation is used
f
(x)
to indicate the derivative of a function with respect to time. For example, the derivative of the
amount of heat transferred, Q, with respect to time, dQ/dt, is often written as
.
Q
It is also of interest to note that many detailed calculations for dynamic systems involve not only
one derivative of a function, but several successive derivatives. The second derivative of a
function is the derivative of its derivative; the third derivative is the derivative of the second
derivative, and so on. For example, velocity is the first derivative of distance traveled with
respect to time, v = ds/dt; acceleration is the derivative of velocity with respect to time, a = dv/dt.
Thus, acceleration is the second derivative of distance traveled with respect to time. This is
written as d2s/dt2. The notation d2[f(x)]/dx2 is the common way to indicate the second derivative
of a function. In some applications, the notation
is used. The notation for third, fourth,
f
(x)
and higher order derivatives follows this same format. Dot notation can also be used for higher
order derivatives with respect to time. Two dots indicates the second derivative, three dots the
third derivative, and so on.
Application of Derivatives to Physical Systems
There are many different problems involving dynamic physical systems that are most readily
solved using derivatives. One of the best illustrations of the application of derivatives are
problems involving related rates of change. When two quantities are related by some known
physical relationship, their rates of change with respect to a third quantity are also related. For
example, the area of a circle is related to its radius by the formula
. If for some reason
A pr2
the size of a circle is changing in time, the rate of change of its area, with respect to time for
example, is also related to the rate of change of its radius with respect to time. Although these
applications involve finding the derivative of function, they illustrate why derivatives are needed
to solve certain problems involving dynamic systems.
MA-05
Page 38
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