Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 14/latex
\setcounter{section}{14}
\zwischenueberschrift{Differentiability}
In this section, we consider functions
\mathdisp {f \colon D \longrightarrow \R} { , }
where
\mavergleichskette
{\vergleichskette
{D
}
{ \subseteq }{\R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
is a subset of the real numbers. We want to explain what it means that such a function is differentiable in a point
\mavergleichskette
{\vergleichskette
{a
}
{ \in }{D
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
The intuitive idea is to look at another point
\mavergleichskette
{\vergleichskette
{x
}
{ \in }{D
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
and to consider the \stichwort {secant,} {} given by the two points
\mathkor {} {(a,f(a))} {and} {(x,f(x))} {,}
and then to let \anfuehrung{$x$ move towards $a$}{.} If this limiting process makes sense, the secants tend to become a tangent. However, this process only has a precise basis, if we use the concept of the limit of a function as defined earlier.
\inputdefinition
{ }
{
Let
\mavergleichskette
{\vergleichskette
{D
}
{ \subseteq }{ \R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
be a subset,
\mavergleichskette
{\vergleichskette
{a
}
{ \in }{D
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
a point, and
\mathdisp {f \colon D \longrightarrow \R} { }
a
function.
For
\mathbed {x \in D} {}
{x \neq a} {}
{} {} {} {,}
the number
\mathdisp {\frac{ f (x )-f (a) }{ x -a }} { }
is called the \definitionswort {difference quotient}{} of $f$ for
}
The difference quotient is the slope of the secant at the graph, running through the two points
\mathkor {} {(a,f(a))} {and} {(x,f(x))} {.}
For
\mavergleichskette
{\vergleichskette
{x
}
{ = }{a
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
this quotient is not defined. However, a useful limit might exist for \mathl{x \rightarrow a}{.} This limit represents, in the case of existence, the slope of the \stichwort {tangent} {} for $f$ in the point \mathl{(a,f(a))}{.}
\inputdefinition
{ }
{
Let
\mavergleichskette
{\vergleichskette
{D
}
{ \subseteq }{ \R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
be a subset,
\mavergleichskette
{\vergleichskette
{a
}
{ \in }{D
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
a point, and
\mathdisp {f \colon D \longrightarrow \R} { }
a
function.
We say that $f$ is \definitionswort {differentiable}{} in $a$ if the
limit
\mathdisp {\operatorname{lim}_{ x \in D \setminus \{ a \} , \, x \rightarrow a } \, \frac{ f (x )-f (a) }{ x -a }} { }
exists. In the case of existence, this limit is called the \definitionswort {derivative}{} of $f$ in $a$, written
\mathdisp {f'(a)} { . }
}
The derivative in a point $a$ is, if it exists, an element in $\R$. Quite often one takes the difference
\mavergleichskette
{\vergleichskette
{h
}
{ \defeq }{ x-a
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
as the parameter for this limiting process, that is, one considers
\mathdisp {\operatorname{lim}_{ h \rightarrow 0 } \, { \frac{ f(a+h)-f(a) }{ h } }} { . }
The condition
\mavergleichskette
{\vergleichskette
{x
}
{ \in }{D \setminus \{a\}
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
translates then to
\mathbed {a+h \in D} {}
{h \neq 0} {}
{} {} {} {.}
If the Function $f$ describes a one-dimensional movement, meaning a time-dependent process on the real line, then the difference quotient \mathl{{ \frac{ f(x)-f(a) }{ x-a } }}{} is the average velocity between the
\zusatzklammer {time} {} {}
points
\mathkor {} {a} {and} {x} {}
and \mathl{f'(a)}{} is the instantaneous velocity in $a$.
\inputbeispiel{}
{
Let
\mavergleichskette
{\vergleichskette
{ s,c
}
{ \in }{ \R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
and let
\mathdisp {\alpha \colon \R \longrightarrow \R
, x \longmapsto sx+c} { , }
be an
affine-linear function.
To determine the
derivative
in a point
\mavergleichskette
{\vergleichskette
{ a
}
{ \in }{ \R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
we consider the difference quotient
\mavergleichskettedisp
{\vergleichskette
{ { \frac{ (sx+c) - (sa+c) }{ x-a } }
}
{ =} { { \frac{ sx - sa }{ x-a } }
}
{ =} { s
}
{ } {
}
{ } {
}
}
{}{}{.}
This is constant and equals $s$, so that the limit of the difference quotient as $x$ tends to $a$ exists and equals $s$ as well. Hence, the derivative exists in every point and is just $s$. The \stichwort {slope} {} of the affine-linear function is also its derivative.
}
\inputbeispiel{}
{
We consider the
function
\mathdisp {f \colon \R \longrightarrow \R
, x \longmapsto x^2} { . }
The
difference quotient
for
\mathkor {} {a} {and} {a+h} {}
is
\mavergleichskettedisp
{\vergleichskette
{ { \frac{ f(a+h) -f(a) }{ h } }
}
{ =} { { \frac{ (a+h)^2-a^2 }{ h } }
}
{ =} { { \frac{ a^2+2ah+h^2 -a^2 }{ h } }
}
{ =} { { \frac{ 2ah+h^2 }{ h } }
}
{ =} { 2a+h
}
}
{}{}{.}
The
limit
of this, as $h$ tends to $0$, is $2a$. The
derivative
of $f$ in $a$ is therefore
\mavergleichskette
{\vergleichskette
{ f'(a)
}
{ = }{ 2a
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
}
\zwischenueberschrift{Linear approximation}
We discuss a property which is equivalent with differentiability, the existence of a linear approximation. This formulation is important in many respects: It allows giving quite simple proofs of the rules for differentiable functions, one can use it to reduce differentiability to the continuity of an error function, it yields a model for approximation with polynomials of higher degree \zusatzklammer {quadratic approximation, Taylor expansion} {} {,} and it allows a direct generalization to the higher-dimensional situation(in the second term)
\inputfaktbeweis
{Differentiable function/D in R/Linear approximation/Fact}
{Theorem}
{}
{
\faktsituation {Let
\mavergleichskette
{\vergleichskette
{D
}
{ \subseteq }{ \R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
be a subset,
\mavergleichskette
{\vergleichskette
{a
}
{ \in }{D
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
a point, and
\mathdisp {f \colon D \longrightarrow \R} { }
a
function.}
\faktfolgerung {Then $f$ is
differentiable
in $a$ if and only if there exists some
\mavergleichskette
{\vergleichskette
{ s
}
{ \in }{ \R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
and a function
\mathdisp {r \colon D \longrightarrow \R} { , }
such that $r$ is
continuous
in $a$,
\mavergleichskette
{\vergleichskette
{r(a)
}
{ = }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
and such that
\mavergleichskettedisp
{\vergleichskette
{ f(x)
}
{ =} { f(a) + s \cdot (x-a) + r(x) (x-a)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}}
\faktzusatz {}
}
{
If $f$ is
differentiable,
then we set
\mavergleichskettedisp
{\vergleichskette
{s
}
{ \defeq} { f'(a)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
Then the only possibility to fulfill the conditions for $r$ is
\mavergleichskettedisp
{\vergleichskette
{ r(x)
}
{ =} { \begin{cases} \frac{ f (x )-f (a) }{ x -a } - s \text{ for } x \neq a\, , \\ 0 \text{ for } x = a \, . \end{cases}
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
Because of differentiability, the limit
\mavergleichskettedisp
{\vergleichskette
{ \operatorname{lim}_{ x \rightarrow a , \, x \in D \setminus \{a\} } r(x)
}
{ =} { \operatorname{lim}_{ x \rightarrow a , \, x \in D \setminus \{a\} } { \left(\frac{ f (x )-f (a) }{ x -a } - s \right) }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
exists, and its value is $0$. This means that $r$ is continuous in $a$.
If
\mathkor {} {s} {and} {r} {}
exist with the described properties, then for
\mavergleichskette
{\vergleichskette
{x
}
{ \neq }{a
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
the relation
\mavergleichskettedisp
{\vergleichskette
{ \frac{ f (x )-f (a) }{ x -a }
}
{ =} { s + r(x)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
holds. Since $r$ is continuous in $a$, the limit on the left-hand side, for \mathl{x \rightarrow a}{,} exists.
The affine-linear function
\mathdisp {D \longrightarrow \R
, x \longmapsto f(a) + f'(a) (x-a)} { , }
is called the \stichwort {affine-linear approximation} {.} The constant function given by the value \mathl{f(a)}{} can be considered as the constant approximation.
\inputfaktbeweis
{Differentiable function/D in R/Continuity in point/Fact}
{Corollary}
{}
{
\faktsituation {Let
\mavergleichskette
{\vergleichskette
{D
}
{ \subseteq }{ \R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
be a subset,
\mavergleichskette
{\vergleichskette
{a
}
{ \in }{D
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
a point, and
\mathdisp {f \colon D \longrightarrow \R} { }
a
function.}
\faktfolgerung {Then $f$ is also continuous in $a$.}
\faktzusatz {}
}
{
This follows immediately from Theorem 14.5 .
\zwischenueberschrift{Rules for differentiable functions}
\inputfaktbeweis
{Differentiable function/D in R/Rules/Fact}
{Lemma}
{}
{
\faktsituation {Let
\mavergleichskette
{\vergleichskette
{D
}
{ \subseteq }{\R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
be a subset,
\mavergleichskette
{\vergleichskette
{a
}
{ \in }{D
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
a point, and
\mathdisp {f,g \colon D \longrightarrow \R} { }
functions
which are
differentiable
in $a$.}
\faktuebergang {Then the following rules for differentiability holds.}
\faktfolgerung {\aufzaehlungfuenf {The sum \mathl{f+g}{} is differentiable in $a$, with
\mavergleichskettedisp
{\vergleichskette
{ (f+g)'(a)
}
{ =} { f'(a) + g'(a)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
} {The product \mathl{f \cdot g}{} is differentiable in $a$, with
\mavergleichskettedisp
{\vergleichskette
{ (f \cdot g)'(a)
}
{ =} { f'(a) g(a) + f(a) g'(a)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
} {For
\mavergleichskette
{\vergleichskette
{ c
}
{ \in }{ \R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
also \mathl{cf}{} is differentiable in $a$, with
\mavergleichskettedisp
{\vergleichskette
{ (cf)'(a)
}
{ =} { c f'(a)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
} {If $g$ has no zero in $a$, then \mathl{1/g}{} is differentiable in $a$, with
\mavergleichskettedisp
{\vergleichskette
{ { \left( { \frac{ 1 }{ g } } \right) }'(a)
}
{ =} { { \frac{ - g'(a) }{ (g(a))^2 } }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
} {If $g$ has no zero in $a$, then \mathl{f/g}{} is differentiable in $a$, with
\mavergleichskettedisp
{\vergleichskette
{ { \left( { \frac{ f }{ g } } \right) }'(a)
}
{ =} { { \frac{ f'(a)g(a) - f(a)g'(a) }{ (g(a))^2 } }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
}}
\faktzusatz {}
}
{
(1). We write
\mathkor {} {f} {and} {g} {}
respectively with the objects which were formulated in
Theorem 14.5
,
that is
\mavergleichskettedisp
{\vergleichskette
{ f(x)
}
{ =} { f(a) + s (x-a) + r(x) (x-a)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
and
\mavergleichskettedisp
{\vergleichskette
{ g(x)
}
{ =} { g(a) + \tilde{s} (x-a) + \tilde{r}(x) (x-a)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
Summing up yields
\mavergleichskettedisp
{\vergleichskette
{ f(x) + g(x)
}
{ =} { f(a) + g(a) + ( s+ \tilde{s} ) (x-a) + (r+ \tilde{r})(x) (x-a)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
Here, the sum \mathl{r+ \tilde{r}}{} is again continuous in $a$, with value $0$.
(2). We start again with
\mavergleichskettedisp
{\vergleichskette
{ f(x)
}
{ =} { f(a) + s (x-a) + r(x) (x-a)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
and
\mavergleichskettedisp
{\vergleichskette
{ g(x)
}
{ =} { g(a) + \tilde{s} (x-a) + \tilde{r}(x) (x-a)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{,}
and multiply both equations. This yields
\mavergleichskettealign
{\vergleichskettealign
{ f(x) g(x)
}
{ =} { ( f(a) + s (x-a) + r(x) (x-a) ) ( g(a) + \tilde{s} (x-a) + \tilde{r}(x) (x-a) )
}
{ =} { f(a)g(a) + ( sg(a) + \tilde{s} f(a)) (x-a)
}
{ \, \, \, \, \,} {+ ( f(a) \tilde{r}(x) + g(a)r(x) + s \tilde{s} (x-a) + s \tilde{r}(x) (x-a) + \tilde{s} r (x) (x-a) + r(x) \tilde{r}(x) (x-a) ) (x-a)
}
{ } {
}
}
{}
{}{.}
Due to
Lemma 10.11
for
limits,
the expression consisting of the last six summands is a continuous function, with value $0$ for
\mavergleichskette
{\vergleichskette
{x
}
{ = }{a
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
(3) follows from (2), since a constant function is differentiable with derivative $0$.
(4). We have
\mavergleichskettedisp
{\vergleichskette
{ \frac{ \frac{1}{g(x)} - \frac{1}{g(a)} }{x-a}
}
{ =} { \frac{-1}{ g(a)g(x)} \cdot \frac{ g (x )-g (a) }{ x -a }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
Since $g$ is continuous in $a$, due to
Corollary 14.6
,
the left-hand factor converges for \mathl{x \rightarrow a}{} to \mathl{- \frac{1}{g(a)^2}}{,} and because of the differentiability of $g$ in $a$, the right-hand factor converges to \mathl{g'(a)}{.}
(5) follows from (2) and (4).
These rules are called \stichwort {sum rule} {,} \stichwort {product rule} {,} \stichwort {quotient rule} {.} The following statement is called \stichwort {chain rule} {.}
\inputfaktbeweis
{Differentiable function/D in R/Chain rule/Fact}
{Theorem}
{}
{
\faktsituation {Let
\mavergleichskette
{\vergleichskette
{D,E
}
{ \subseteq }{ \R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
denote subsets, and let
\mathdisp {f \colon D \longrightarrow \R} { }
and
\mathdisp {g \colon E \longrightarrow \R} { }
be functions with
\mavergleichskette
{\vergleichskette
{ f(D)
}
{ \subseteq }{ E
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}}
\faktvoraussetzung {Suppose that $f$ is
differentiable
in $a$ and that $g$ is differentiable in
\mavergleichskette
{\vergleichskette
{ b
}
{ \defeq }{ f(a)
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}}
\faktfolgerung {Then also the
composition
\mathdisp {g \circ f \colon D \longrightarrow \R} { }
is differentiable in $a$, and its
derivative
is
\mavergleichskettedisp
{\vergleichskette
{ ( g \circ f)' (a)
}
{ =} { g'(f(a)) \cdot f'(a)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}}
\faktzusatz {}
}
{
Due to
Theorem 14.5
,
one can write
\mavergleichskettedisp
{\vergleichskette
{ f(x)
}
{ =} { f ( a) + f' ( a) ( x - a ) + r(x) ( x - a)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
and
\mavergleichskettedisp
{\vergleichskette
{ g(y)
}
{ =} { g ( b) + g' ( b) ( y - b ) + s(y) ( y - b)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
Therefore,
\mavergleichskettealign
{\vergleichskettealign
{ g(f(x))
}
{ =} { g ( f(a)) + g' ( f(a) ) ( f(x) - f(a) ) + s(f(x)) ( f(x) - f(a))
}
{ =} { g(f(a)) +g'(f(a)) { \left( f'(a)(x-a) +r(x)(x-a) \right) } +s(f(x)) { \left( f'(a)(x-a) +r(x)(x-a) \right) }
}
{ =} { g(f(a)) +g'(f(a)) f'(a)(x-a) + { \left( g'(f(a)) r(x) + s(f(x)) (f'(a) +r(x) ) \right) } (x-a)
}
{ } {
}
}
{}
{}{.}
The remainder function
\mavergleichskettedisp
{\vergleichskette
{ t(x)
}
{ \defeq} { g'(f(a)) r(x) + s(f(x)) (f'(a) +r(x) )
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
is continuous in $a$ with value $0$.
\inputfaktbeweis
{Differentiable function/D in R/Inverse function/Fact}
{Theorem}
{}
{
\faktsituation {Let
\mavergleichskette
{\vergleichskette
{D,E
}
{ \subseteq }{ \R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
denote
intervals,
and let
\mathdisp {f \colon D \longrightarrow E \subseteq \R} { }
be a
bijective
continuous function,
with the
inverse function
\mathdisp {f^{-1} \colon E \longrightarrow D} { . }
}
\faktvoraussetzung {Suppose that $f$ is
differentiable
in
\mavergleichskette
{\vergleichskette
{a
}
{ \in }{D
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
with
\mavergleichskette
{\vergleichskette
{ f'(a)
}
{ \neq }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}}
\faktfolgerung {Then also the
inverse function
$f^{-1}$ is differentiable in
\mavergleichskette
{\vergleichskette
{b
}
{ \defeq }{f(a)
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
and
\mavergleichskettedisp
{\vergleichskette
{ (f^{-1})'(b)
}
{ =} { { \frac{ 1 }{ f' (f^{-1} (b)) } }
}
{ =} { { \frac{ 1 }{ f'(a) } }
}
{ } {
}
{ } {
}
}
{}{}{}
holds.}
\faktzusatz {}
}
{
We consider the difference quotient
\mavergleichskettedisp
{\vergleichskette
{ \frac{f^{-1} (y) - f^{-1} (b) }{y-b}
}
{ =} { \frac{f^{-1} (y) -a }{y-b}
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{,}
and have to show that the limit for \mathl{y \rightarrow b}{} exists, and obtains the value claimed. For this, let \mathl{{ \left( y_n \right) }_{n \in \N }}{} denote a
sequence
in \mathl{E \setminus \{b\}}{,}
converging
to $b$. Because of
Theorem 11.7
,
the function $f^{-1}$ is continuous. Therefore, also the sequence with the members
\mavergleichskette
{\vergleichskette
{ x_n
}
{ \defeq }{ f^{-1}(y_n)
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
converges to $a$. Because of bijectivity,
\mavergleichskette
{\vergleichskette
{x_n
}
{ \neq }{a
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
for all $n$. Thus
\mavergleichskettedisp
{\vergleichskette
{ \lim_{ n \rightarrow \infty} \frac{ f^{-1}(y_n) -a }{ y_n - b }
}
{ =} { \lim_{ n \rightarrow \infty} \frac{ x_n -a }{ f(x_n) - f(a) }
}
{ =} { { \left( \lim_{ n \rightarrow \infty} \frac{ f(x_n) - f(a) }{x_n -a}\right) }^{-1}
}
{ } {
}
{ } {
}
}
{}{}{,}
where the right-hand side exists, due to the condition, and the second equation follows from
Lemma 8.1
.
\inputbeispiel{}
{
The function
\mathdisp {f^{-1} \colon \R_+ \longrightarrow \R_+
, x \longmapsto \sqrt{x}} { , }
is the
inverse function
of the function $f$, given by
\mavergleichskette
{\vergleichskette
{ f(x)
}
{ = }{x^2
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
\zusatzklammer {restricted to $\R_+$} {} {.}
The
derivative
of $f$ in a point $a$ is
\mavergleichskette
{\vergleichskette
{f'(a)
}
{ = }{2a
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Due to
Theorem 14.9
,
for
\mavergleichskette
{\vergleichskette
{b
}
{ \in }{\R_+
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
the relation
\mavergleichskettedisp
{\vergleichskette
{ { \left( f^{-1} \right) }' (b)
}
{ =} { \frac{1}{f'(f^{-1} (b))}
}
{ =} { \frac{1}{2 \sqrt{b} }
}
{ =} { \frac{1}{2} b^{-\frac{1}{2} }
}
{ } {
}
}
{}{}{}
holds. In the zero point, however, $f^{-1}$ is not differentiable.
}
\inputbeispiel{}
{
The function
\mathdisp {f^{-1} \colon \R \longrightarrow \R
, x \longmapsto x^{\frac{1}{3} }} { , }
is the
inverse function
of the function $f$, given by
\mavergleichskette
{\vergleichskette
{f(x)
}
{ = }{x^3
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
The derivative of $f$ in $a$ is
\mavergleichskette
{\vergleichskette
{f'(a)
}
{ = }{ 3a^2
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
which is different from $0$ for
\mavergleichskette
{\vergleichskette
{a
}
{ \neq }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
Due to
Theorem 14.9
,
we have for
\mavergleichskette
{\vergleichskette
{b
}
{ \neq }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
the relation
\mavergleichskettedisp
{\vergleichskette
{ { \left( f^{-1} \right) }' (b)
}
{ =} { \frac{1}{f'(f^{-1} (b))}
}
{ =} { \frac{1}{3 { \left( b^{\frac{1}{3} } \right) }^{2} }
}
{ =} { \frac{1}{3} b^{-\frac{2}{3} }
}
{ } {
}
}
{}{}{.}
In the zero point, however, $f^{-1}$ is not differentiable.
}
\zwischenueberschrift{The derivative function}
So far, we have considered differentiability of a function in just one point, now we consider the derivative in general.
\inputdefinition
{ }
{
Let
\mavergleichskette
{\vergleichskette
{ I }
{ \subseteq }{ \R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
denote an
interval,
and let
\mathdisp {f \colon I \longrightarrow \R} { }
be a
function.
We say that $f$ is \definitionswort {differentiable}{,} if for every point
\mavergleichskette
{\vergleichskette
{ a
}
{ \in }{ I
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
the
derivative
\mathl{f'(a)}{} of $f$ in $a$ exists. In this case, the
mapping
\mathdisp {f' \colon I \longrightarrow \R
, x \longmapsto f'(x)} { , }
}