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Definiere + Funktion + \[ + R(x) := f(x) - f(x_0) - c(x - x_0) + .\] Dann gilt + \[ + \frac{R(x)}{x- x_0} = \frac{f(x) - f(x_0)}{x - x_0} - \underbrace{c}_{f'(x_0)} + \xrightarrow[x \to x_0]{f \text{ diff.}} 0 + .\] + + ,,$\impliedby$ '' Sei umgekehrt $c \in \R$ mit + \[ + \frac{R(x)}{x - x_0} = \frac{f(x) - f(x_0)}{x - x_0} - c \xrightarrow{x \to x_0} 0 + .\], d.h. + \[ + \lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0} = c + .\] $\implies f'(x_0) = c$. Limes eindeutig $\implies$ $f$ differenzierbar. +\end{proof} + +\begin{bem} + Aus dem Satz zur linearen Approximation folgt eine geometrische Interpretation: $f(x)$ + kann in $x_0$ ,,gut'' durch eine Gerade approximiert werden. + \[ + f(x) \approx g(x) = f(x_0) + f'(x_0)(x- x_0) + .\] Der Graph von $g$ ist eine Tangente. + Sekante: + \[ + s_h(x) = f(x_0) + \frac{f(x_0 + h) - f(x_0)}{h} (x-x_0) + .\] Tangente: + \[ + g(x) = f(x_0) + f'(x)(x-x_0) + .\] +\end{bem} +\begin{figure}[htpb] + \centering + \caption{$f(x)$ in rot, ihre Tangente (blau) und eine Sekante (lila) + im Punkt $x_0 = 1$} + \begin{tikzpicture} + \begin{axis}% + [grid=both, + minor tick num=4, + grid style={line width=.1pt, draw=gray!10}, + major grid style={line width=.2pt,draw=gray!50}, + axis lines=middle, + enlargelimits={abs=0.2}, + ymax=5, + ymin=0 + ] + \addplot[domain=0:3,samples=50,smooth,red] {0.5*x^2}; + \addplot[domain=0:3,samples=50,smooth,blue] {(x - 1)+0.5}; + \addplot[domain=0:3,samples=50,smooth,purple] {1.9*(x - 1) + 0.5}; + \end{axis} + \end{tikzpicture} +\end{figure} + +\begin{lemma} + Sei $f\colon D \to \R$ differenzierbar in $x_0 \in D$. Dann ist + $f$ stetig in $x_0$. +\end{lemma} + +\begin{proof} + Sei $f$ differenzierbar, d.h. + \[ + \exists f'(x_0) = \lim_{x \to x_0} \frac{f(x) - f(x_0)}{x - x_0} + .\] Dann gilt wegen der linearen Approximation: + \begin{align*} + f(x) &= f(x_0) + f'(x_0)(x - x_0) + R(x) \\ + &= f(x_0) + f'(x_0)(x-x_0) + \frac{R(x)}{x - x_0}(x-x_0) + .\end{align*} + Für $x \to x_0$ geht + \[ + f(x) - f(x_0) = f'(x_0)\underbrace{(x-x_0)}_{\to 0} + + \underbrace{\frac{R(x)}{x -x_0}}_{\to 0}\underbrace{(x-x_0)}_{\to 0} + .\] d.h. $f(x) \to f(x_0) \stackrel{\text{Def. Stetigkeit}}{\implies} f$ stetig. +\end{proof} + +\begin{bem} + Umgekehrt gilt das nicht, z.B.: die Betragsfunktion. +\end{bem} + +\begin{bsp} + \begin{enumerate} + \item Konstante Funktionen $f \equiv c$ sind stetig + differenzierbar mit $f'(x_0) = 0$ $\forall x_0$. + \item Lineare Funktionen $f\colon \R \to \R$ + $f = ax$ sind stetig differenzierbar mit $f'(x_0) = a$ + $\forall x_0$, weil + \[ + \lim_{h \to 0} \frac{a(x_0 + h) - ax_0}{h} = a + .\] + \item Monomfunktion: $f(x) = x^{n}, n \in \N$ ist + stetig differenzierbar mit $f'(x) = n x^{n-1}$ $\forall x$, weil + \begin{align*} + \lim_{h \to 0} \frac{(x+h)^{n} - x^{n}}{h} + &\stackrel{a^{n} - b^{n} = \ldots}{=} + \lim_{h \to 0} \frac{(x+h-x)(x+h)^{n-1}+(x+h)^{n-2}\cdot x + \ldots + (x-h)x^{n-2} + x^{n-1}}{h} \\ + &= \underbrace{x^{n-1} + x^{n-2} \cdot x + \ldots + x^{n-1}}_{n\text{-mal}} \\ + &= n x^{n-1} + .\end{align*} + \item Elementare rationale Funktionen + $f = \frac{1}{x}$, $x \neq 0$. + \begin{align*} + f'(x) &= \lim_{h \to 0} \frac{1}{h} \left( \frac{1}{x+h} - \frac{1}{x} \right) + = \lim_{h \to 0} \frac{1}{h} \frac{x - (x+h)}{(x+h)\cdot x} + = \lim_{h \to 0} - \frac{1}{\underbrace{(x+h)}_{\to x}\cdot x} + = - \frac{1}{x^2} + .\end{align*} + \item Betragsfunktion $f(x) = |x|$ + \[ + f'(x) = \begin{cases} + x & x \ge 0 \\ + -x & x < 0 + \end{cases} + .\] ist bei $x_0 = 0$ nicht differenzierbar. + $\frac{d|x|}{dx}$ für $x_0 = 0$ existiert nicht. Allerdings + existieren die einseitigen Ableitungen. +\begin{figure}[htpb] + \centering + \caption{Betragsfunktion} + \begin{tikzpicture} + \begin{axis}% + [grid=both, + minor tick num=4, + grid style={line width=.1pt, draw=gray!10}, + major grid style={line width=.2pt,draw=gray!50}, + axis lines=middle, + enlargelimits={abs=0.2}, + ymax=5, + ymin=0 + ] + \addplot[domain=-3:3,samples=100,red] {abs(x)}; + \end{axis} + \end{tikzpicture} +\end{figure} + \item Exponential-Funktion $f(x) = e^{x}$ ist stetig + differenzierbar $\forall x$ mit $f'(x) = e^{x}$, weil + \begin{align*} + \lim_{h \to 0} \frac{e^{x+h} - e^{x}}{h} + = e^{x} \cdot \underbrace{\lim_{h \to 0} \frac{e^{h}-1}{h}}_{= 1} = e^{x} + .\end{align*} + mit + \begin{align*} + e^{h} &= \sum_{k=0}^{\infty} \frac{h^{k}}{k!} = 1 + \frac{h}{2} + \frac{h^2}{3!} + \ldots \\ + \frac{e^{h} - 1}{h} &= 1 + \frac{h}{2} + \frac{h^2}{3!} + \ldots \xrightarrow{h \to 0} \to 1 + .\end{align*} + \item Sinus / Cosinus. + mit $\sin(x) - \sin(y) = 2 \cos \frac{1}{2} (x+y) \cdot \sin \frac{1}{2}(x-y)$ folgt + \begin{align*} + \sin'(x) &= \lim_{h \to 0} \frac{\sin(x+h) - \sin(x)}{h} \\ + &= \lim_{h \to 0} \frac{2 \cos\left(\frac{1}{2} h +x\right)\cdot \sin(\frac{1}{2}h)}{h} \\ + &= \lim_{h \to 0} + \underbrace{\cos\left(\frac{1}{2}h + x\right)}_{\to \cos x} + \cdot + \underbrace{\lim_{h \to 0} \frac{\sin(\frac{1}{2}h)}{\frac{h}{2}}}_{\to 1} \\ + &= \cos x + .\end{align*} + + $\cos'(x) = - \sin(x)$ folgt analog. + \end{enumerate} +\end{bsp} + +\begin{satz}[Ableitungsregeln] + Für die Ableitungen gelten folgende Rechenregeln. + Seien $f, g\colon D \to \R$ differenzierbar. + \begin{enumerate} + \item Lineare Kombinationen $\alpha f + \beta g$ + ist differenzierbar mit + \[ + (\alpha f + \beta g)'(x) = \alpha f'(x) + \beta g'(x) + .\] $\alpha, \beta \in \R$ + \item Produktregel + \[ + (f\cdot g)'(x) = f'(x)g(x) + f(x)\cdot g'(x) + .\] + \item Quotientenregel + \[ + \left(\frac{f}{g}\right)'(x) = \frac{f'(x) \cdot g(x) - f(x)\cdot g'(x)}{g^2} + .\] $g(x) \neq 0$ + \end{enumerate} +\end{satz} + +\begin{satz}[Differenzierbarkeit der Umkehrfunktion] + Sei $f\colon D \to B \subset \R$ stetige invertierbare + Funktion mit Inverser + \[ + f^{-1}\colon B \to D. + .\] Ist $f$ in $x_0 \in D$ differenzierbar mit $f'(x) \neq 0$. Dann ist + $f^{-1}$ in $y_0 = f(x_0)$ differenzierbar mit + \[ + \left( f^{-1} \right)'(y_0) = \frac{1}{f'(x_0)} \quad y_0 = f(x_0) + .\] +\end{satz} + +\begin{satz}[Kettenregel] + Seien $f\colon D_f \to \R$, $g\colon D_g \to \R$ stetige Funktionen. + $f \in x_0 \in D_f$ differenzierbar, $g \in y_0 = f(x_0) \in D_g$ + differenzierbar. Dann ist $(g \circ f): D_f \to \R$ differenzierbar + in $x_0$ und es gilt die Kettenregel + \[ + \left( g \circ f \right) ' (x_0) = g'(f(x_0)) \cdot f'(x_0) + .\] +\end{satz} + +\begin{bsp} + Für $x > 0$ \[ + \frac{d}{dx}(\ln x) = \frac{1}{x} + .\] $\ln x$ auf $]0, \infty[$ ist stetig differenzierbar. + \[ + \ln'(y) = \frac{1}{(e^{x})'} = \frac{1}{e^{x}} = \frac{1}{y} + .\] $y = e^{x}$ + + Trick: $y = u^{v}$, $u = u(x), v = v(x)$ + \begin{align*} + \ln y &= v \ln u \\ + \frac{1}{y} \cdot y' &= v' \ln u + v \cdot \ln + v\cdot (\ln u)' + = v' \ln u + v \frac{1}{u} u' \\ + \implies y' &= y (v' \ln u + v \frac{1}{u} u') \\ + \implies (u^{v})' &= u^{v}(v' \ln u + v \cdot \frac{1}{u} u') + = u^{v} \cdot \ln u \cdot v' + u^{v-1} \cdot v \cdot u' + .\end{align*} + + $y = \frac{(x^2 + 2)\cdot \sqrt[4]{(x-1)^{3}} e^{x} }{(x+5)^{3}} = g(x)$ \\ + $\ln y = \ln(x^2 + 2) + \frac{3}{4} (x-1) + x - 3 \ln (x+5)$ +\end{bsp} + +\end{document}