130
D. Rocchesso: Sound Processing
composition of functions
The derivative is a linear operator, i.e.,
· The derivative of a sum of functions is the sum of the derivatives of the
single functions
· The derivative of a product of a function by a constant is the product of
the constant by the derivative of the function
Another important property of the derivative is that it transforms the com-
position of functions in a product of functions. Given two functions y = f (x)
and z = g(y), the composed function z = g(f (x)) is obtained by replacing
the domain of the second function with the codomain of the first one
derivative of the composed function is expressed as
dz
dx
= g (y)f (x) =
dz
dy
dy
dx
,
(35)
which remarks the effectiveness of the notation introduced for the derivatives.
For the purpose of this book, it is useful to know the derivatives of the main
trigonometric functions, which are given by
d sin x
dx
= cos x
(36)
d cos x
dx
= - sin x
(37)
d tan x
dx
=
1
cos
2
x
(38)
Therefore, we can say that a sinusoidal function conserves its sinusoidal charac-
ter (it is only translated along the x axis) when it is subject to derivation. This
property comes from the fact, already anticipated, that the exponential with
base e is an eigenfunction for the derivative operator, i.e.,
de
x
dx
= e
x
.
(39)
If we consider the complex exponential e
ix
as the composition of an exponential
function with a monomial with imaginary coefficient, it is possible to apply the
linearity of derivative to the composed function and derive the formulas (36)
and (37).
In order to derive (38) we also have to know the rule to derive quotients of
functions. In general, products and quotients of functions are derived according
to
d [f (x)g(x)]
dx
= f (x)g(x) + f (x)g (x)
(40)
d [g(x)/f (x)]
dx
=
g (x)f (x) - f (x)g(x)
f
2
(x)
.
(41)
12
For instance, log x
2
is obtained by squaring x and then taking the logarithm or, by the
property L3 of logarithms, ...