# Mathematics for Applied Sciences (Osnabrück 2023-2024)/Part I/Lecture 16/latex

\setcounter{section}{16}
\zwischenueberschrift{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.
\faktsituation {Let \mavergleichskettedisp {\vergleichskette {g(x) } { \defeq} { \sum _{ n= 0}^\infty a_n x^{ n } } { } { } { } { } { } { } } denote a power series} \faktvoraussetzung {which converges on the open interval \mathl- r,r{,} and represents there a function $f \colon ]-r,r[ \rightarrow \R$.} \faktfolgerung {Then the formally differentiated power series \mavergleichskettedisp {\vergleichskette { \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 \mavergleichskettedisp {\vergleichskette { f'(x) } { =} { \tilde{ g}(x) } { } { } { } { } { } { } } holds.} \faktzusatz } {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.
\faktsituation {A function given by a power series} \faktfolgerung {is infinitely often differentiable on its interval of convergence.} \faktzusatz } {
This follows immediately from Theorem 16.1 .
\faktsituation {The exponential function \mathdisp {\R \longrightarrow \R , x \longmapsto \exp x} { , } } \faktfolgerung {is differentiable with \mavergleichskettedisp {\vergleichskette { \exp \!'( x ) } { =} { \exp x } { } { } { } { } { } { } }{.}} \faktzusatz } {
Due to Theorem 16.1 , we have \mavergleichskettealign {\vergleichskettealign { \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} } } { \vergleichskettefortsetzungalign { =} { \sum_{ n =0}^\infty \frac{ x^{ n } }{n!} } { =} { \exp x } { } { } }{.}
\faktsituation {The exponential function \mathdisp {\R \longrightarrow \R , x \longmapsto a^x} { , } with base \mavergleichskette {\vergleichskette {a } { > }{0 } { }{ } { }{ } { }{ } }{,}} \faktfolgerung {is differentiable with \mavergleichskettedisp {\vergleichskette { { \left( a^x \right) }' } { =} { { \left( \ln a \right) } a^x } { } { } { } { } { } { } }{.}} \faktzusatz } {
By definition, we have \mavergleichskettedisp {\vergleichskette { a^x } { =} { \exp { \left( x \, \ln a \right) } } { } { } { } { } { } { } }{.} The derivative with respect to $x$ equals \mavergleichskettedisp {\vergleichskette { { \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 \mavergleichskettedisp {\vergleichskette {y(x) } { =} {a^x } { } { } { } { } { } { } }{,} the relation \mavergleichskettedisp {\vergleichskette {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 \mavergleichskette {\vergleichskette {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 \zusatzklammer {or decreasing behavior, if \mavergleichskette {\vergleichskette {a } { < }{1 } { }{ } { }{ } { }{ } }} occurs in nature for a population, if there is no competition for resources, and if the dying rate is neglectable \zusatzklammer {the number of mice is then proportional with the number of mice born} {.} A condition of the form \mavergleichskettedisp {\vergleichskette {y' } { =} {b y } { } { } { } { } { } { } } is an example of a \stichwort {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 \mavergleichskettedisp {\vergleichskette {y(x) } { =} {ce^{bx} } { } { } { } { } { } { } }{.}
We will study differential equations in the second semester.
\faktsituation \faktvoraussetzung {The derivative of the natural logarithm \mathdisp {\ln \colon \R_+ \longrightarrow \R , x \longmapsto \ln x} { , } } \faktfolgerung {is \mathdisp {\ln \!' \colon \R_+ \longrightarrow \R , x \longmapsto \frac{1}{x}} { . } } \faktzusatz } {
As the logarithm is the inverse function of the exponential function, we can apply Theorem 14.9
and get \mavergleichskettedisp {\vergleichskette { \ln' (x) } { =} { { \frac{ 1 }{ \exp' ( \ln x) } } } { =} { { \frac{ 1 }{ \exp ( \ln x) } } } { =} { { \frac{ 1 }{ x } } } { } { } }{,} using Theorem 16.3 .
\faktsituation {Let \mavergleichskette {\vergleichskette { \alpha } { \in }{ \R } { }{ } { }{ } { }{ } }{.}} \faktfolgerung {Then the function \mathdisp {f \colon \R_+ \longrightarrow \R_+ , x \longmapsto x^\alpha} { , } is differentiable, and its derivative is \mavergleichskettedisp {\vergleichskette { f'(x) } { =} { \alpha x^{\alpha -1} } { } { } { } { } { } { } }{.}} \faktzusatz } {
By definition, we have \mavergleichskettedisp {\vergleichskette { x^\alpha } { =} { \exp { \left( \alpha \, \ln x \right) } } { } { } { } { } { } { } }{.} The derivative with respect to $x$ equals \mavergleichskettedisp {\vergleichskette { { \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.
\faktsituation \faktfolgerung {The sine function \mathdisp {\R \longrightarrow \R , x \longmapsto \sin x} { , } is differentiable, with \mavergleichskettedisp {\vergleichskette { \sin \!'( x) } { =} { \cos x } { } { } { } { } { } { } }{,} and the cosine function \mathdisp {\R \longrightarrow \R , x \longmapsto \cos x} { , } is differentiable, with \mavergleichskettedisp {\vergleichskette { \cos \!'( x ) } { =} { - \sin x } { } { } { } { } { } { } }{.}} \faktzusatz
\faktsituation \faktfolgerung {The tangent function \mathdisp {\R \setminus { \left({ \frac{ \pi }{ 2 } } + \Z \pi\right) } \longrightarrow \R , x \longmapsto \tan x} { , } is differentiable, with \mavergleichskettedisp {\vergleichskette { \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 \mavergleichskettedisp {\vergleichskette { \cot \!'( x ) } { =} { - { \frac{ 1 }{ \sin^{ 2 } x } } } { } { } { } { } { } { } }{.}} \faktzusatz } {
Using the quotient rule, Theorem 16.8 , and the circle equation, we get \mavergleichskettealign {\vergleichskettealign {(\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.
\zwischenueberschrift{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.
\faktsituation \faktvoraussetzung {The cosine function} \faktfolgerung {has, within the real interval \mathl{[0,2]}{,} exactly one zero.} \faktzusatz } {
We consider the cosine series \mavergleichskettedisp {\vergleichskette { \cos x } { =} { \sum_{ n = 0}^\infty \frac{ (-1)^{ n } x^{2n} }{(2n)!} } { } { } { } { } { } { } }{.} For \mavergleichskette {\vergleichskette {x } { = }{ 0 } { }{ } { }{ } { }{ } }{,} we have \mavergleichskette {\vergleichskette { \cos 0 } { = }{ 1 } { }{ } { }{ } { }{ } }{.} For \mavergleichskette {\vergleichskette {x } { = }{2 } { }{ } { }{ } { }{ } }{,} one can write \mavergleichskettealign {\vergleichskettealign { \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 \mavergleichskettedisp {\vergleichskette { \cos ' x } { =} { - \sin x } { } { } { } { } { } { } }{,} due to Theorem 16.8 . Hence, it is enough to show that sine is positive in the interval \mathl0,2{,} because then cosine is strictly decreasing by Theorem 15.7
in the interval and there is only one zero. Now, for \mavergleichskette {\vergleichskette {x } { \in }{0,2]} } { }{ } { }{ } { }{ } }{,} we have \mavergleichskettealign {\vergleichskettealign { \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 } } { \vergleichskettefortsetzungalign { >} {0 } { } { } { } }{.}
\inputdefinition { } {
Let $s$ denote the unique \zusatzklammer {according to Lemma 16.10 } real zero of the cosine function in the interval \mathl{[0,2]}{.} Then the number $\pi$ is defined by \mavergleichskettedisp {\vergleichskette { \pi } { \defeq} { 2s } { } { } { } { } { } { } }
\faktsituation {The sine function and the cosine function fulfill in $\R$ the following periodicity properties.} \faktfolgerung {\aufzaehlungfuenf {We have \mavergleichskette {\vergleichskette { \cos { \left( x +2 \pi \right) } } { = }{ \cos x } { }{ } { }{ } { }{ } } and \mavergleichskette {\vergleichskette { \sin { \left( x +2 \pi \right) } } { = }{ \sin x
} { }{ } { }{ } { }{ } } for all \mavergleichskette {\vergleichskette { x } { \in }{ \R } { }{ } { }{ } { }{ } }{.} } {We have \mavergleichskette {\vergleichskette { \cos { \left( x + \pi \right) } } { = }{ - \cos x } { }{ } { }{ } { }{ } } and \mavergleichskette {\vergleichskette { \sin { \left( x + \pi \right) } } { = }{ - \sin x } { }{ } { }{ } { }{ } } for all \mavergleichskette {\vergleichskette { x } { \in }{ \R } { }{ } { }{ } { }{ } }{.} } {We have \mavergleichskette {\vergleichskette { \cos { \left( x + \pi/2 \right) } } { = }{ - \sin x } { }{ } { }{ } { }{ } } and \mavergleichskette {\vergleichskette { \sin { \left( x + \pi/2 \right) } } { = }{ \cos x } { }{ } { }{ } { }{ } } for all \mavergleichskette {\vergleichskette { x } { \in }{ \R } { }{ } { }{ } { }{ } }{.} } {We have \mavergleichskette {\vergleichskette { \cos 0 } { = }{ 1 } { }{ } { }{ } { }{ } }{,} \mavergleichskette {\vergleichskette { \cos \pi/2 } { = }{ 0 } { }{ } { }{ } { }{ } }{,} \mavergleichskette {\vergleichskette { \cos \pi } { = }{ -1 } { }{ } { }{ } { }{ } }{,} \mavergleichskette {\vergleichskette { \cos 3\pi/2 } { = }{ 0 } { }{ } { }{ } { }{ } }{,} and \mavergleichskette {\vergleichskette { \cos 2 \pi } { = }{ 1 } { }{ } { }{ } { }{ } }{.} } {We have \mavergleichskette {\vergleichskette { \sin 0 } { = }{ 0 } { }{ } { }{ } { }{ } }{,} \mavergleichskette {\vergleichskette { \sin \pi/2 } { = }{1 } { }{ } { }{ } { }{ } }{,} \mavergleichskette {\vergleichskette { \sin \pi } { = }{ 0 } { }{ } { }{ } { }{ } }{,} \mavergleichskette {\vergleichskette { \sin 3\pi/2 } { = }{ -1 } { }{ } { }{ } { }{ } }{,} and \mavergleichskette {\vergleichskette { \sin 2 \pi } { = }{ 0 } { }{ } { }{ } { }{ } }{.} }} \faktzusatz } {
Due to the circle equation \mavergleichskettedisp {\vergleichskette { (\cos z )^2 + ( \sin z)^2 } { =} { 1 } { } { } { } { } { } { } }{,} we have \mavergleichskette {\vergleichskette { { \left( \sin \frac{\pi}{2} \right) }^2 } { = }{ 1 } { }{ } { }{ } { }{ } }{,} hence \mavergleichskette {\vergleichskette { \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. \mavergleichskettedisp {\vergleichskette { \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 \definitionswort {periodic} with \definitionswort {period length} \mavergleichskette {\vergleichskette {L } { > }{0 } { }{ } { }{ } { }{ } }{,} if the equality \mavergleichskettedisp {\vergleichskette { f(x) } { =} { f(x+L) } { } { } { } { } { } { } } holds for all \mavergleichskette {\vergleichskette {x } { \in }{\R } { }{ } { }{ } { }{ } }
The trigonometric functions sin and cosine are periodic functions with the period length $2 \pi$.
\zwischenueberschrift{The inverse trigonometric functions}
\faktsituation {The real sine function} \faktfolgerung {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]} { . } } \faktzusatz
\faktsituation \faktfolgerung {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} { . } } \faktzusatz
Due to the bijectivity of sine, cosine, tangent and cotangent on suitable interval, there exist the following inverse functions.
\inputdefinition { } {
The inverse function of the real sine function is \mathdisp {[-1,1] \longrightarrow [- \frac{\pi}{2}, \frac{\pi}{2}] , x \longmapsto \arcsin x} { , }
\inputdefinition { } {
The inverse function of the real cosine function is \mathdisp {[-1,1] \longrightarrow [0, \pi] , x \longmapsto \arccos x} { , }
\inputdefinition { } {
The inverse function of the real tangent function is \mathdisp {\R \longrightarrow ] - { \frac{ \pi }{ 2 } } , { \frac{ \pi }{ 2 } } [ , x \longmapsto \arctan x} { , }
\inputdefinition { } {
The inverse function of the real cotangent function is \mathdisp {\R \longrightarrow ] 0 , \pi [ , x \longmapsto \arccot x} { , }
\faktsituation {The inverse trigonometric functions have the following derivatives.} \faktfolgerung {\aufzaehlungvier { \mavergleichskettedisp {\vergleichskette { { \left( \arcsin x \right) }' } { =} { { \frac{ 1 }{ \sqrt{1-x^2} } } } { } { } { } { } { } { } }{.} } { \mavergleichskettedisp {\vergleichskette { { \left( \arccos x \right) }' } { =} { - { \frac{ 1 }{ \sqrt{1-x^2} } } } { } { } { } { } { } { } }{.} } { \mavergleichskettedisp {\vergleichskette { { \left( \arctan x \right) }' } { =} { { \frac{ 1 }{ 1+x^2 } } } { } { } { } { } { } { } }{.} } { \mavergleichskettedisp {\vergleichskette { { \left( \arccot x \right) }' } { =} {- { \frac{ 1 }{ 1+x^2 } } } { } { } { } { } { } { } }{.} }} \faktzusatz
For example, for the arctangent, we have, due to Theorem 14.9 , \mavergleichskettealign {\vergleichskettealign { (\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 } } } }{.}