Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 16/latex
\setcounter{section}{16}
\subtitle {Derivative of power series}
Many important functions, like the exponential function or the trigonometric functions, are represented by a power series. The following theorem shows that these functions are differentiable, and that the derivative of a power series is itself a power series, given by differentiating the individual terms of the series.
\inputfaktbeweis
{Real power series/Derivative by formal derivative/Fact}
{Theorem}
{}
{
\factsituation {Let
\mathrelationchaindisplay
{\relationchain
{g(x)
}
{ \defeq} { \sum _{ n= 0}^\infty a_n x^{ n }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
denote a
power series}
\factcondition {which converges
on the
open interval
\mathl{]- r,r[}{,} and represents there a function
$f \colon ]-r,r[ \rightarrow \R$.}
\factconclusion {Then the formally differentiated power series
\mathrelationchaindisplay
{\relationchain
{ \tilde{g}(x)
}
{ \defeq} { \sum_{n = 1}^\infty n a_n x^{n-1}
}
{ } {
}
{ } {}
{ } {}
}
{}{}{}
is convergent on \mathl{]-r,r[}{.} The function $f$ is
differentiable
in every point of the interval, and
\mathrelationchaindisplay
{\relationchain
{ f'(x)
}
{ =} { \tilde{ g}(x)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
holds.}
\factextra {}
}
{Real power series/Derivative by formal derivative/Fact/Proof
In the formulation of the theorem, we have distinguished between $g$ for the power series and $f$ for the function, defined by the series, in order to stress the roles they play. This distinction is now not necessary anymore.
\inputfactproof
{Real power series/Infinitely often differentiable/Fact}
{Corollary}
{}
{
\factsituation {A function given by a power series}
\factconclusion {is infinitely often differentiable on its interval of convergence.}
\factextra {}
}
{
This follows immediately from Theorem 16.1 .
\inputfactproof
{Real exponential function/Derivative/Fact}
{Theorem}
{}
{
\factsituation {The
exponential function
\mathdisp {\R \longrightarrow \R
, x \longmapsto \exp x} { , }
}
\factconclusion {is
differentiable
with
\mathrelationchaindisplay
{\relationchain
{ \exp \!'( x )
}
{ =} { \exp x
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}}
\factextra {}
}
{
Due to
Theorem 16.1
,
we have
\mathrelationchainalign
{\relationchainalign
{ \exp \!'( x)
}
{ =} { { \left(\sum_{ n =0}^\infty \frac{ x^{ n } }{n!}\right) }'
}
{ =} { \sum_{n = 1 }^\infty { \left(\frac{ x^n}{n !}\right) }'
}
{ =} { \sum_{n = 1 }^\infty \frac{n }{n !} x^{n-1}
}
{ =} { \sum_{n = 1 }^\infty \frac{1 }{(n-1) !}x^{n-1}
}
}
{
\relationchainextensionalign
{ =} { \sum_{ n =0}^\infty \frac{ x^{ n } }{n!}
}
{ =} { \exp x
}
{ } {}
{ } {}
}
{}{.}
\inputfactproof
{Real exponential function/Base/Derivative/Fact}
{Theorem}
{}
{
\factsituation {The
exponential function
\mathdisp {\R \longrightarrow \R
, x \longmapsto a^x} { , }
with base
\mathrelationchain
{\relationchain
{a
}
{ > }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}}
\factconclusion {is
differentiable
with
\mathrelationchaindisplay
{\relationchain
{ { \left( a^x \right) }'
}
{ =} { { \left( \ln a \right) } a^x
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}}
\factextra {}
}
{
By
definition,
we have
\mathrelationchaindisplay
{\relationchain
{ a^x
}
{ =} { \exp { \left( x \, \ln a \right) }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
The
derivative
with respect to $x$ equals
\mathrelationchaindisplay
{\relationchain
{ { \left( a^x \right) }'
}
{ =} { { \left( \exp { \left( x \, \ln a \right) } \right) }'
}
{ =} { { \left( \ln a \right) } \exp' (x \, \ln a )
}
{ =} { { \left( \ln a \right) } \exp { \left( x \, \ln a \right) }
}
{ =} { { \left( \ln a \right) } a^x
}
}
{}{}{,}
due to
Theorem 16.3
and
the chain rule.
\inputremark {}
{
For a
real exponential function
\mathrelationchaindisplay
{\relationchain
{y(x)
}
{ =} {a^x
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{,}
the relation
\mathrelationchaindisplay
{\relationchain
{y'
}
{ =} { { \left( \ln a \right) } y
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
holds, due to
Theorem 16.4
.
Hence, there is a proportional relationship between the function $y$ and its derivative $y'$, and
$\ln a$ is the factor. This is still true if $a^x$ is multiplied with a constant. If we consider $y$ as a function depending on time $x$, then \mathl{y'(x)}{} describes the growing behavior at that point of time. The equation
\mathrelationchain
{\relationchain
{y'
}
{ = }{ { \left( \ln a \right) } y
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
means that the instantaneous growing rate is always proportional with the magnitude of the function. Such an increasing behavior
\extrabracket {or decreasing behavior, if
\mathrelationchain
{\relationchain
{a
}
{ < }{1
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}} {} {}
occurs in nature for a population, if there is no competition for resources, and if the dying rate is neglectable
\extrabracket {the number of mice is then proportional with the number of mice born} {} {.}
A condition of the form
\mathrelationchaindisplay
{\relationchain
{y'
}
{ =} {b y
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
is an example of a \keyword {differential equation} {.} This is an equation for a function, which expresses a condition for the derivative. A solution for such a differential equation is a differentiable function which fulfills the condition on its derivative. The differential equation just mentioned are fulfilled by the functions
\mathrelationchaindisplay
{\relationchain
{y(x)
}
{ =} {ce^{bx}
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
}
We will study differential equations in the second semester.
\inputfactproof
{Natural logarithm/Derivative/Fact}
{Corollary}
{}
{
\factsituation {}
\factcondition {The
derivative
of the
natural logarithm
\mathdisp {\ln \colon \R_+ \longrightarrow \R
, x \longmapsto \ln x} { , }
}
\factconclusion {is
\mathdisp {\ln \!' \colon \R_+ \longrightarrow \R
, x \longmapsto \frac{1}{x}} { . }
}
\factextra {}
}
{
As the logarithm is the inverse function of the exponential function, we can apply
Theorem 14.9
and get
\mathrelationchaindisplay
{\relationchain
{ \ln' (x)
}
{ =} { { \frac{ 1 }{ \exp' ( \ln x) } }
}
{ =} { { \frac{ 1 }{ \exp ( \ln x) } }
}
{ =} { { \frac{ 1 }{ x } }
}
{ } {
}
}
{}{}{,}
using
Theorem 16.3
.
\inputfactproof
{Power function/Positive base/Real exponent/Fact}
{Corollary}
{}
{
\factsituation {Let
\mathrelationchain
{\relationchain
{ \alpha
}
{ \in }{ \R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}}
\factconclusion {Then the
function
\mathdisp {f \colon \R_+ \longrightarrow \R_+
, x \longmapsto x^\alpha} { , }
is
differentiable,
and its
derivative
is
\mathrelationchaindisplay
{\relationchain
{ f'(x)
}
{ =} { \alpha x^{\alpha -1}
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}}
\factextra {}
}
{
By
definition,
we have
\mathrelationchaindisplay
{\relationchain
{ x^\alpha
}
{ =} { \exp { \left( \alpha \, \ln x \right) }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
The
derivative
with respect to $x$ equals
\mathrelationchaindisplay
{\relationchain
{ { \left( x^\alpha \right) }'
}
{ =} { { \left( \exp { \left( \alpha \, \ln x \right) } \right) }'
}
{ =} { \frac{\alpha}{x} \cdot \exp { \left( \alpha\, \ln x \right) }
}
{ =} { \frac{\alpha}{x} x^\alpha
}
{ =} { \alpha x^{\alpha -1}
}
}
{}{}{}
using
Theorem 16.3
,
Corollary 16.6
and
the chain rule.
\inputfactproof
{Real sine and cosine function/Derivative/Fact}
{Theorem}
{}
{
\factsituation {}
\factconclusion {The
sine function
\mathdisp {\R \longrightarrow \R
, x \longmapsto \sin x} { , }
is
differentiable,
with
\mathrelationchaindisplay
{\relationchain
{ \sin \!'( x)
}
{ =} { \cos x
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{,}
and the
cosine function
\mathdisp {\R \longrightarrow \R
, x \longmapsto \cos x} { , }
is differentiable, with
\mathrelationchaindisplay
{\relationchain
{ \cos \!'( x )
}
{ =} { - \sin x
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}}
\factextra {}
{See Exercise 16.4 .}
\inputfactproof
{Tangent/Cotangent/Derivative/Fact}
{Theorem}
{}
{
\factsituation {}
\factconclusion {The
tangent function
\mathdisp {\R \setminus { \left({ \frac{ \pi }{ 2 } } + \Z \pi\right) } \longrightarrow \R
, x \longmapsto \tan x} { , }
is
differentiable,
with
\mathrelationchaindisplay
{\relationchain
{ \tan \!'( x)
}
{ =} { { \frac{ 1 }{ \cos^{ 2 } x } }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{,}
and the
cotangent function
\mathdisp {\R \setminus \Z \pi \longrightarrow \R
, x \longmapsto \cot x} { , }
is differentiable, with
\mathrelationchaindisplay
{\relationchain
{ \cot \!'( x )
}
{ =} { - { \frac{ 1 }{ \sin^{ 2 } x } }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}}
\factextra {}
}
{
Using
the quotient rule,
Theorem 16.8
,
and
the circle equation,
we get
\mathrelationchainalign
{\relationchainalign
{(\tan x )^\prime
}
{ =} { { \left( { \frac{ \sin x }{ \cos x } } \right) }^\prime
}
{ =} { { \frac{ (\cos x)( \cos x ) - ( \sin x )(- \sin x ) }{ \cos^{ 2 } x } }
}
{ =} { { \frac{ 1 }{ \cos^{ 2 } x } }
}
{ } {}
}
{}
{}{.}
The derivative of the cotangent function follows in the same way.
\subtitle {The number $\pi$ }
The number $\pi$ is the area and half of the circumference of a circle with radius $1$. But, in order to build a precise definition for this number on this, we would have first to establish measure theory or the theory of the length of curves. Also, the trigonometric functions have an intuitive interpretation at the unit circle, but also this requires the concept of the arc length. An alternative approach is to define the functions sine and cosine by their power series, and then to define the number $\pi$ with the help of them, and establishing finally the relation with the circle.
\inputfactproof
{Real cosine function/One zero between 0 and 2/Fact}
{Lemma}
{}
{
\factsituation {}
\factcondition {The
cosine function}
\factconclusion {has, within the
real interval
\mathl{[0,2]}{,} exactly one
zero.}
\factextra {}
}
{
We consider the
cosine series
\mathrelationchaindisplay
{\relationchain
{ \cos x
}
{ =} { \sum_{ n = 0}^\infty \frac{ (-1)^{ n } x^{2n} }{(2n)!}
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
For
\mathrelationchain
{\relationchain
{x
}
{ = }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
we have
\mathrelationchain
{\relationchain
{ \cos 0
}
{ = }{ 1
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
For
\mathrelationchain
{\relationchain
{x
}
{ = }{2
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
one can write
\mathrelationchainalign
{\relationchainalign
{ \cos 2
}
{ =} { 1- \frac{2^2}{2!} + \frac{2^4}{4!} - \frac{2^6}{6!} + \frac{2^8}{8!} - \ldots
}
{ =} { 1- \frac{2^2}{2!} { \left( 1 - \frac{ 4}{3 \cdot 4} \right) } - \frac{2^6}{6!} { \left( 1- \frac{4}{7 \cdot 8} \right) } - \ldots
}
{ =} { 1 - 2 ( 2/3) - \ldots
}
{ \leq} { - 1/3
}
}
{}
{}{.}
Hence, due to
the intermediate value theorem,
there exists at least one zero in the given interval.
To prove uniqueness, we consider the
derivative
of cosine, which is
\mathrelationchaindisplay
{\relationchain
{ \cos ' x
}
{ =} { - \sin x
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{,}
due to
Theorem 16.8
.
Hence, it is enough to show that sine is positive in the interval \mathl{]0,2[}{,} because then cosine is
strictly decreasing
by
Theorem 15.7
in the interval and there is only one zero. Now, for
\mathrelationchain
{\relationchain
{x
}
{ \in }{ {]0,2]}
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
we have
\mathrelationchainalign
{\relationchainalign
{ \sin x
}
{ =} { x- \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \ldots
}
{ =} { x { \left( 1- \frac{x^2}{3!} \right) } + \frac{x^5}{5!} { \left( 1- \frac{x^2}{6 \cdot 7} \right) } + \ldots
}
{ \geq} { x { \left( 1- \frac{4}{3!} \right) } + \frac{x^5}{5!} { \left( 1- \frac{4}{6 \cdot 7} \right) } + \ldots
}
{ \geq} { x/3
}
}
{
\relationchainextensionalign
{ >} {0
}
{ } {}
{ } {}
{ } {}
}
{}{.}
\image{ \begin{center}
\includegraphics[width=5.5cm]{\imageinclude {Pi pie2.jpg} }
\end{center}
\imagetext {A rational approximation of the number $\pi$ on a $\pi$-pie.} }
\imagelicense { Pi pie2.jpg } {} {GJ} {engl. Wikipedia} {PD} {}
\inputdefinition
{ }
{
Let $s$ denote the unique
\extrabracket {according to
Lemma 16.10
} {} {}
real
zero
of the
cosine function
in the
interval
\mathl{[0,2]}{.} Then the number $\pi$ is defined by
\mathrelationchaindisplay
{\relationchain
{ \pi
}
{ \defeq} { 2s
}
{ } {
}
{ } {
}
{ } {
}
}
}
\inputfactproof
{Sine and cosine/R/Properties of periodicity/Fact}
{Theorem}
{}
{
\factsituation {The sine function and the cosine function fulfill in $\R$ the following periodicity properties.}
\factconclusion {\enumerationfive {We have
\mathrelationchain
{\relationchain
{ \cos { \left( x +2 \pi \right) }
}
{ = }{ \cos x
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
and
\mathrelationchain
{\relationchain
{ \sin { \left( x +2 \pi \right) }
}
{ = }{ \sin x
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
for all
\mathrelationchain
{\relationchain
{ x
}
{ \in }{ \R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
} {We have
\mathrelationchain
{\relationchain
{ \cos { \left( x + \pi \right) }
}
{ = }{ - \cos x
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
and
\mathrelationchain
{\relationchain
{ \sin { \left( x + \pi \right) }
}
{ = }{ - \sin x
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
for all
\mathrelationchain
{\relationchain
{ x
}
{ \in }{ \R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
} {We have
\mathrelationchain
{\relationchain
{ \cos { \left( x + \pi/2 \right) }
}
{ = }{ - \sin x
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
and
\mathrelationchain
{\relationchain
{ \sin { \left( x + \pi/2 \right) }
}
{ = }{ \cos x
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{}
for all
\mathrelationchain
{\relationchain
{ x
}
{ \in }{ \R
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
} {We have
\mathrelationchain
{\relationchain
{ \cos 0
}
{ = }{ 1
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
\mathrelationchain
{\relationchain
{ \cos \pi/2
}
{ = }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
\mathrelationchain
{\relationchain
{ \cos \pi
}
{ = }{ -1
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
\mathrelationchain
{\relationchain
{ \cos 3\pi/2
}
{ = }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
and
\mathrelationchain
{\relationchain
{ \cos 2 \pi
}
{ = }{ 1
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
} {We have
\mathrelationchain
{\relationchain
{ \sin 0
}
{ = }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
\mathrelationchain
{\relationchain
{ \sin \pi/2
}
{ = }{1
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
\mathrelationchain
{\relationchain
{ \sin \pi
}
{ = }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
\mathrelationchain
{\relationchain
{ \sin 3\pi/2
}
{ = }{ -1
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
and
\mathrelationchain
{\relationchain
{ \sin 2 \pi
}
{ = }{ 0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{.}
}}
\factextra {}
}
{
Due to the
circle equation
\mathrelationchaindisplay
{\relationchain
{ (\cos z )^2 + ( \sin z)^2
}
{ =} { 1
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{,}
we have
\mathrelationchain
{\relationchain
{ { \left( \sin \frac{\pi}{2} \right) }^2
}
{ = }{ 1
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
hence
\mathrelationchain
{\relationchain
{ \sin \frac{\pi}{2}
}
{ = }{ 1
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
because of the reasoning in the proof of
Lemma 16.10
.
From that we deduce, with the help of
the addition theorems,
the relations between sine and cosine as mentioned in (3), e.g.
\mathrelationchaindisplay
{\relationchain
{ \cos { \left( z + { \frac{ \pi }{ 2 } } \right) }
}
{ =} { \cos z \cos { \left( { \frac{ \pi }{ 2 } } \right) } - \sin z \sin { \left( { \frac{ \pi }{ 2 } } \right) }
}
{ =} { - \sin z
}
{ } {
}
{ } {
}
}
{}{}{.}
Hence it is enough to prove the statements for cosine. All statements follow from the definition of $\pi$ and from (3).
\inputdefinition
{ }
{
A
function
$f \colon \R \rightarrow \R$
is called \definitionword {periodic}{} with \definitionword {period length}{}
\mathrelationchain
{\relationchain
{L
}
{ > }{0
}
{ }{
}
{ }{
}
{ }{
}
}
{}{}{,}
if the equality
\mathrelationchaindisplay
{\relationchain
{ f(x)
}
{ =} { f(x+L)
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{}
holds for all
\mathrelationchain
{\relationchain
{x
}
{ \in }{\R
}
{ }{
}
{ }{
}
{ }{
}
}
}
The trigonometric functions sin and cosine are periodic functions with the period length $2 \pi$.
\subtitle {The inverse trigonometric functions}
\inputfactproof
{Sine and cosine/Monotonicity behavior/Fact}
{Corollary}
{}
{
\factsituation {The
real sine function}
\factconclusion {induces a
bijective,
strictly increasing
function
\mathdisp {[- \pi/2, \pi/2] \longrightarrow [-1,1]} { , }
and the
real cosine function
induces a bijective, strictly decreasing function
\mathdisp {[0,\pi] \longrightarrow [-1,1]} { . }
}
\factextra {}
{See Exercise 16.13 .}
\inputfactproof
{Tangent and cotangent/Monotonicity behavior/Fact}
{Corollary}
{}
{
\factsituation {}
\factconclusion {The
real tangent function
induces a
bijective,
strictly increasing
function
\mathdisp {]- \pi/2, \pi/2[ \longrightarrow \R} { , }
and the
real cotangent function
induces a bijective strictly decreasing function
\mathdisp {[0,\pi] \longrightarrow \R} { . }
}
\factextra {}
{See Exercise 16.14 .}
Due to the bijectivity of sine, cosine, tangent and cotangent on suitable interval, there exist the following inverse functions.
\image{ \begin{center}
\includegraphics[width=5.5cm]{\imageinclude {Arcsine.svg} }
\end{center}
\imagetext {} }
\imagelicense { Arcsine.svg } {} {Geek3} {Commons} {CC-by-sa 4.0} {}
\inputdefinition
{ }
{
The
inverse function
of the real
sine function
is
\mathdisp {[-1,1] \longrightarrow [- \frac{\pi}{2}, \frac{\pi}{2}]
, x \longmapsto \arcsin x} { , }
}
\image{ \begin{center}
\includegraphics[width=5.5cm]{\imageinclude {Arccosine.svg} }
\end{center}
\imagetext {} }
\imagelicense { Arccosine.svg } {} {Geek3} {Commons} {CC-by-sa 4.0} {}
\inputdefinition
{ }
{
The
inverse function
of the real
cosine function
is
\mathdisp {[-1,1] \longrightarrow [0, \pi]
, x \longmapsto \arccos x} { , }
}
\image{ \begin{center}
\includegraphics[width=5.5cm]{\imageinclude {Arctangent.svg} }
\end{center}
\imagetext {Arkustangens} }
\imagelicense { Arctangent.svg } {} {Geek3} {Commons} {CC-by-sa 4.0} {}
\inputdefinition
{ }
{
The
inverse function
of the real
tangent function
is
\mathdisp {\R \longrightarrow ] - { \frac{ \pi }{ 2 } } , { \frac{ \pi }{ 2 } } [
, x \longmapsto \arctan x} { , }
}
\image{ \begin{center}
\includegraphics[width=5.5cm]{\imageinclude {Arccotangent.svg} }
\end{center}
\imagetext {} }
\imagelicense { Arccotangent.svg } {} {Geek3} {Commons} {CC-by-sa 4.0} {}
\inputdefinition
{ }
{
The
inverse function
of the real
cotangent function
is
\mathdisp {\R \longrightarrow ] 0 , \pi [
, x \longmapsto \arccot x} { , }
}
\inputfactproof
{Inverse trigonometric functions/Derivative/Fact}
{Theorem}
{}
{
\factsituation {The inverse trigonometric functions have the following
derivatives.}
\factconclusion {\enumerationfour {
\mathrelationchaindisplay
{\relationchain
{ { \left( \arcsin x \right) }'
}
{ =} { { \frac{ 1 }{ \sqrt{1-x^2} } }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
} {
\mathrelationchaindisplay
{\relationchain
{ { \left( \arccos x \right) }'
}
{ =} { - { \frac{ 1 }{ \sqrt{1-x^2} } }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
} {
\mathrelationchaindisplay
{\relationchain
{ { \left( \arctan x \right) }'
}
{ =} { { \frac{ 1 }{ 1+x^2 } }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
} {
\mathrelationchaindisplay
{\relationchain
{ { \left( \arccot x \right) }'
}
{ =} {- { \frac{ 1 }{ 1+x^2 } }
}
{ } {
}
{ } {
}
{ } {
}
}
{}{}{.}
}}
\factextra {}
}
{
For example, for the arctangent, we have, due to
Theorem 14.9
,
\mathrelationchainalign
{\relationchainalign
{ (\arctan x )^\prime
}
{ =} { { \frac{ 1 }{ { \frac{ 1 }{ \cos^{ 2 } (\arctan x) } } } }
}
{ =} { { \frac{ 1 }{ { \frac{ \cos^{ 2 } (\arctan x) + \sin^{ 2 } (\arctan x) }{ \cos^{ 2 } (\arctan x) } } } }
}
{ =} { { \frac{ 1 }{ 1 + \tan^{ 2 } (\arctan x ) } }
}
{ =} { { \frac{ 1 }{ 1 + x^2 } }
}
}
{}
{}{.}