596
B Review of Traditional Calculus
d
dx
F (x) =
d
dx
f (g(x))
=
df
dg
·
dg
dx
≡ f
g · g
x .
(B.1.2)
(iii) Another rule that occurs with some frequency is the quotient rule, which may
be recalled in the rather complicated, but explicit, form
d
dx
f (x)
g(x)
=
g(x)f
x − f (x)g
x
g 2 (x)
.
(B.1.3)
It may, however, be obtained fairly straightforwardly as a special case of the
product rule taken together with the quotient rule in which F (x) is given by
F (x) = 1/g(x), so that the quotient f/g is considered as the product f F of
the functions f and F .
(iv) Derivatives of non-algebraic functions: a number of simple non-algebraic
functions that are encountered in typical statistical mechanical problems are
the exponential, natural logarithmic, trigonometric, and hyperbolic functions,
for which the most relevant derivatives are summarized below.
d
dx
e
x
= e
x ,
d
dx
e
f (x)
= e
f (x) df
dx
,
d
dx
ln x =
1
x
,
d
dx
ln f (x) =
1
f (x)
df
dx
d
dx
sin x = cos x ,
d
dx
cos x = − sin x ,
d
dx
sinh x = cosh x ,
d
dx
cosh x = sinh x .
B.1.2 Multivariate Functions
(i) Definition of Partial Derivatives. Consider a (continuous) function of the two
independent variables x and y. We shall firstly establish the defining relations
for what is called a partial derivative by employing an explicit form, z =
f (x, y), for the manner in which z depends on the two independent variables
x and y even though, more often than not, the dependence of z upon x and y
can only be provided implicitly via an expression of the form f (x, y, z) = 0.
For an implicit relation, we shall see that it is a matter of choice as to which
of the three variables involved in the expression is to be considered to be the
dependent variable.
B Review of Traditional Calculus
d
dx
F (x) =
d
dx
f (g(x))
=
df
dg
·
dg
dx
≡ f
g · g
x .
(B.1.2)
(iii) Another rule that occurs with some frequency is the quotient rule, which may
be recalled in the rather complicated, but explicit, form
d
dx
f (x)
g(x)
=
g(x)f
x − f (x)g
x
g 2 (x)
.
(B.1.3)
It may, however, be obtained fairly straightforwardly as a special case of the
product rule taken together with the quotient rule in which F (x) is given by
F (x) = 1/g(x), so that the quotient f/g is considered as the product f F of
the functions f and F .
(iv) Derivatives of non-algebraic functions: a number of simple non-algebraic
functions that are encountered in typical statistical mechanical problems are
the exponential, natural logarithmic, trigonometric, and hyperbolic functions,
for which the most relevant derivatives are summarized below.
d
dx
e
x
= e
x ,
d
dx
e
f (x)
= e
f (x) df
dx
,
d
dx
ln x =
1
x
,
d
dx
ln f (x) =
1
f (x)
df
dx
d
dx
sin x = cos x ,
d
dx
cos x = − sin x ,
d
dx
sinh x = cosh x ,
d
dx
cosh x = sinh x .
B.1.2 Multivariate Functions
(i) Definition of Partial Derivatives. Consider a (continuous) function of the two
independent variables x and y. We shall firstly establish the defining relations
for what is called a partial derivative by employing an explicit form, z =
f (x, y), for the manner in which z depends on the two independent variables
x and y even though, more often than not, the dependence of z upon x and y
can only be provided implicitly via an expression of the form f (x, y, z) = 0.
For an implicit relation, we shall see that it is a matter of choice as to which
of the three variables involved in the expression is to be considered to be the
dependent variable.