Differentiable applications
Définition : Differentiable applications
Let
be an application defined on an open interval
of
with values in
Let
and
be two elements of
We call growth rate of
between
and
the quotient
We say that application
is differentiable in
if the growth rate has a finite limit when
goes towards
The real number
is called derivative number of application
in
and denoted by
If the growth rate has a rigth-sided limit when
goes to
we say that application
is differentiable from the rigth in
We define similarly the applications that are differentiable from the left.
Geometric interpretation
The derivative number
of a differentiable application in point
is the slope of the tangent line of the graphical representation of application
in the point of coordinates
Derivative application
If an application
is differentiable in any point of an interval
we define an application from
to
denoted by
and called derivative application of application
Fondamental : Sum, produic, quotient and composite of differentiable applications
Let
be an interval of
and
two differentiable applications on
Application
is differentiable and
'
Application
is differentiable and
If application
does not take the zero value on interval
application
is differentiable and
Let
and
be two intervals of
a differentiable application from
to
and
an application from
to
Application
is differentiable on
and
Differentiability of the inverse of a bijective application
Let
be a bijective differentiable application defined on an interval
of
and with values in an interval
Application
is differentiable in a point
belonging to
if and only if
is non zero and we have:
The derivative application of application
is
Fondamental : Derivative and direction of variation of an application
Let
be a differentiable application on an interval
of
If application
is positive on interval
application
is increasing on
If application
is negative on interval
application
is decreasing on
Définition : Derivatives of upper orders
Let
be a differentiable application on a interval
with values in
If application
is continuous on
we say that application
is of class
on
If application
is differentiable, we denote by
its derivative, called second derivative of function
If the second derivative of
is continuous, we say that
is of class
We can define in a recursive way the derivative of order
of
which will be called of class
if it has a continuous derivative of order
If function
is of class
for any natural integer
we say it is of class
Fondamental : Rolle's theorem, formula of mean value
Rolle's theorem
Let
be a continuous application on a segment
of
differentiable on
such that
then:
There exists a point
such that
Formula of mean value
Let
be a continuous application on
and differentiable on
There exists
[ such that
