# 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
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)
\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.
\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}
\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} {.}
\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 $0$.
\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)} { , }