Appendix B
Review of Traditional Calculus
B.1 Derivatives
B.1.1 Functions of a Single Variable
Statistical thermodynamics and statistical mechanics typically involve the use of
derivatives, both of functions of a single variable and of multivariable functions.
In the latter case, we shall typically not examine functions of more than two
independent variables, as all of the important properties that will be needed can
be illustrated in terms of functions of three variables. We begin simply by listing
some of the most important (or, for our purposes, useful) results taught in typical
first year calculus courses.
(i) One of the most important rules is the product rule, which gives the derivative
of the product of two functions of a single variable in terms of products of these
functions and their derivatives. It takes the form
d
dx
{f (x)g(x)} =
df
dx
g(x) + f (x)
dg
dx
≡ f
x g(x) + f (x) g
x .
(B.1.1)
The latter version employs the shorthand notation f
x for the derivative of f (x)
with respect to x.
(ii) Another very important rule of differentiation is the chain rule for obtaining
the derivative of a composite function F (x) = f (g(x)) in terms of simpler
derivatives. This rule may be expressed as
© Springer Nature Switzerland AG 2021
F. R. W. McCourt, Statistical Thermodynamics for Pure and Applied Sciences,
https://doi.org/10.1007/978-3-030-52006-9
595
Review of Traditional Calculus
B.1 Derivatives
B.1.1 Functions of a Single Variable
Statistical thermodynamics and statistical mechanics typically involve the use of
derivatives, both of functions of a single variable and of multivariable functions.
In the latter case, we shall typically not examine functions of more than two
independent variables, as all of the important properties that will be needed can
be illustrated in terms of functions of three variables. We begin simply by listing
some of the most important (or, for our purposes, useful) results taught in typical
first year calculus courses.
(i) One of the most important rules is the product rule, which gives the derivative
of the product of two functions of a single variable in terms of products of these
functions and their derivatives. It takes the form
d
dx
{f (x)g(x)} =
df
dx
g(x) + f (x)
dg
dx
≡ f
x g(x) + f (x) g
x .
(B.1.1)
The latter version employs the shorthand notation f
x for the derivative of f (x)
with respect to x.
(ii) Another very important rule of differentiation is the chain rule for obtaining
the derivative of a composite function F (x) = f (g(x)) in terms of simpler
derivatives. This rule may be expressed as
© Springer Nature Switzerland AG 2021
F. R. W. McCourt, Statistical Thermodynamics for Pure and Applied Sciences,
https://doi.org/10.1007/978-3-030-52006-9
595