Lösung zur Ableitung der Formel
Eingabefunktion
$$\frac{\ln\left(x^{2} + 1\right)}{\ln\left(x\right)}$$
Start der Ableitung
$$\frac{d}{dx}{\left(\frac{\log\left(x^{2} + 1\right)}{\log\left(x\right)}\right)}$$
Schritt 1 — Quotientenregel
Regel
$$\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{u'v-uv'}{v^{2}}$$
Mit:
- $u = \log\left(x^{2} + 1\right)$
- $v = \log\left(x\right)$
Aktueller Ausdruck
$$\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left(\frac{\log\left(x^{2} + 1\right)}{\log\left(x\right)}\right)}\,}}}$$
Nach Quotientenregel
$$\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{\frac{d}{dx}{\left(\log\left(x^{2} + 1\right)\right)}\log\left(x\right) - \log\left(x^{2} + 1\right)\frac{d}{dx}{\left(\log\left(x\right)\right)}}{\left(\log\left(x\right)\right)^{2}}\,}}}$$
Schritt 2 — Kettenregel
Regel
$$\frac{d}{dx}f(u)=f'(u)\cdot u'$$
Mit:
- $u = x^{2} + 1$
Aktueller Ausdruck
$$\frac{\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left(\log\left(x^{2} + 1\right)\right)}\,}}}\log\left(x\right) - \log\left(x^{2} + 1\right)\frac{d}{dx}{\left(\log\left(x\right)\right)}}{\left(\log\left(x\right)\right)^{2}}$$
Nach Kettenregel
$$\frac{\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{x^{2} + 1}\,\frac{d}{dx}{\left(x^{2} + 1\right)}\,}}}\log\left(x\right) - \log\left(x^{2} + 1\right)\frac{d}{dx}{\left(\log\left(x\right)\right)}}{\left(\log\left(x\right)\right)^{2}}$$
Schritt 3 — Summen-/Differenzenregel
Regel
$$\frac{d}{dx}(u \pm v \pm \dots) = \frac{d}{dx}(u) \pm \frac{d}{dx}(v) \pm \dots$$
Mit:
- $u = x^{2}$
- $v = 1$
Aktueller Ausdruck
$$\frac{\frac{1}{x^{2} + 1}\,\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left(x^{2} + 1\right)}\,}}}\log\left(x\right) - \log\left(x^{2} + 1\right)\frac{d}{dx}{\left(\log\left(x\right)\right)}}{\left(\log\left(x\right)\right)^{2}}$$
Nach Summen-/Differenzenregel
$$\frac{\frac{1}{x^{2} + 1} \cdot \left(\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left(x^{2}\right)} + \frac{d}{dx}{\left(1\right)}\,}}}\right)\log\left(x\right) - \log\left(x^{2} + 1\right)\frac{d}{dx}{\left(\log\left(x\right)\right)}}{\left(\log\left(x\right)\right)^{2}}$$
Schritt 4 — Potenzregel (Spezialfall)
Regel
$$\frac{d}{dx}\left(u^{c}\right)=c\,u^{c-1}$$
Mit:
- $u = x$
- $c = 2$
Aktueller Ausdruck
$$\frac{\frac{1}{x^{2} + 1} \cdot \left(\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left(x^{2}\right)}\,}}} + \frac{d}{dx}{\left(1\right)}\right)\log\left(x\right) - \log\left(x^{2} + 1\right)\frac{d}{dx}{\left(\log\left(x\right)\right)}}{\left(\log\left(x\right)\right)^{2}}$$
Nach Potenzregel (Spezialfall)
$$\frac{\frac{1}{x^{2} + 1} \cdot \left(\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,x^{2 - 1}\,}}} + \frac{d}{dx}{\left(1\right)}\right)\log\left(x\right) - \log\left(x^{2} + 1\right)\frac{d}{dx}{\left(\log\left(x\right)\right)}}{\left(\log\left(x\right)\right)^{2}}$$
Vereinfacht
$$\frac{\frac{1}{x^{2} + 1} \cdot \left(\bbox[lightgreen, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,x\,}}} + \frac{d}{dx}{\left(1\right)}\right)\log\left(x\right) - \log\left(x^{2} + 1\right)\frac{d}{dx}{\left(\log\left(x\right)\right)}}{\left(\log\left(x\right)\right)^{2}}$$
Schritt 5 — Konstantenregel
Regel
$$\frac{d}{dx}(c)=0$$
Mit:
- $c = 1$
Aktueller Ausdruck
$$\frac{\frac{1}{x^{2} + 1} \cdot \left(2\,x + \bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left(1\right)}\,}}}\right)\log\left(x\right) - \log\left(x^{2} + 1\right)\frac{d}{dx}{\left(\log\left(x\right)\right)}}{\left(\log\left(x\right)\right)^{2}}$$
Nach Konstantenregel
$$\frac{\frac{1}{x^{2} + 1} \cdot \left(2\,x + \bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,0\,}}}\right)\log\left(x\right) - \log\left(x^{2} + 1\right)\frac{d}{dx}{\left(\log\left(x\right)\right)}}{\left(\log\left(x\right)\right)^{2}}$$
Vereinfacht
$$\frac{\bbox[lightgreen, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{x^{2} + 1} \cdot 2\,x\,}}}\log\left(x\right) - \log\left(x^{2} + 1\right)\frac{d}{dx}{\left(\log\left(x\right)\right)}}{\left(\log\left(x\right)\right)^{2}}$$
Schritt 6 — Kettenregel
Regel
$$\frac{d}{dx}f(u)=f'(u)\cdot u'$$
Mit:
- $u = x$
Aktueller Ausdruck
$$\frac{\frac{1}{x^{2} + 1} \cdot 2\,x\log\left(x\right) - \log\left(x^{2} + 1\right)\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left(\log\left(x\right)\right)}\,}}}}{\left(\log\left(x\right)\right)^{2}}$$
Nach Kettenregel
$$\frac{\frac{1}{x^{2} + 1} \cdot 2\,x\log\left(x\right) - \log\left(x^{2} + 1\right) \cdot \bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{x}\,}}}}{\left(\log\left(x\right)\right)^{2}}$$
Ergebnis
$$\frac{\frac{1}{x^{2} + 1} \cdot 2\,x\log\left(x\right) - \log\left(x^{2} + 1\right) \cdot \frac{1}{x}}{\left(\log\left(x\right)\right)^{2}}$$
Direkt berechnet (Maxima)
Ableitung der Eingabefunktion via Maxima:
$$\frac{2\,x}{\left(x^{2} + 1\right)\,\log\left(x\right)} - \frac{\log\left(x^{2} + 1\right)}{x\left(\log\left(x\right)\right)^{2}}$$
Abgleich Schrittfolge ↔︎ Maxima:
gleich (true)