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
$$\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{\frac{d}{dx}{\left[\log\left(x^{2} + 1\right)\right]}\,\log\left(x\right) - \log\left(x^{2} + 1\right)\,\frac{d}{dx}{\left[\log\left(x\right)\right]}}{\log\left(x\right)^{2}}\,}}}$$
Schritt 2 — Kettenregel für den Logarithmus
📖 Regel
$$\frac{d}{dx}\big(\log({u})\big) = \frac{u'}{{u}}$$
Mit:
- $u = x^{2} + 1$
🧮 Aktueller Ausdruck
$$\frac{\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\log\left(x^{2} + 1\right)\right]}\,}}}\,\log\left(x\right) - \log\left(x^{2} + 1\right)\,\frac{d}{dx}{\left[\log\left(x\right)\right]}}{\log\left(x\right)^{2}}$$
✨ Nach Kettenregel für den Logarithmus
$$\frac{\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{x^{2} + 1}\,\frac{d}{dx}{\left[x^{2} + 1\right]}\,}}}\,\log\left(x\right) - \log\left(x^{2} + 1\right)\,\frac{d}{dx}{\left[\log\left(x\right)\right]}}{\log\left(x\right)^{2}}$$
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
$$\frac{\frac{1}{x^{2} + 1} \cdot \bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x^{2} + 1\right]}\,}}}\,\log\left(x\right) - \log\left(x^{2} + 1\right)\,\frac{d}{dx}{\left[\log\left(x\right)\right]}}{\log\left(x\right)^{2}}$$
✨ Nach Summen-/Differenzenregel
$$\frac{\frac{1}{x^{2} + 1} \cdot \bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x^{2}\right]} + \frac{d}{dx}{\left[1\right]}\,}}}\,\log\left(x\right) - \log\left(x^{2} + 1\right)\,\frac{d}{dx}{\left[\log\left(x\right)\right]}}{\log\left(x\right)^{2}}$$
Schritt 4 — Potenzregel (Spezialfall)
📖 Regel
$$\frac{d}{dx}\left(u^{c}\right)=c\,u^{c-1}$$
Mit:
- $u = x$
- $c = 2$
🧮 Aktueller Ausdruck
$$\frac{\frac{1}{x^{2} + 1} \cdot \left(\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x^{2}\right]}\,}}} + \frac{d}{dx}{\left[1\right]}\right)\,\log\left(x\right) - \log\left(x^{2} + 1\right)\,\frac{d}{dx}{\left[\log\left(x\right)\right]}}{\log\left(x\right)^{2}}$$
✨ Nach Potenzregel (Spezialfall)
$$\frac{\frac{1}{x^{2} + 1} \cdot \left(\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2\,x^{2 - 1}\,}}} + \frac{d}{dx}{\left[1\right]}\right)\,\log\left(x\right) - \log\left(x^{2} + 1\right)\,\frac{d}{dx}{\left[\log\left(x\right)\right]}}{\log\left(x\right)^{2}}$$
🧹 Vereinfacht
$$\frac{\frac{1}{x^{2} + 1} \cdot \left(2\,x^{\bbox[lightgreen, padding:0.12em 0.22em]{\vphantom{\dfrac{d}{dx}}\,1\,}} + \frac{d}{dx}{\left[1\right]}\right)\,\log\left(x\right) - \log\left(x^{2} + 1\right)\,\frac{d}{dx}{\left[\log\left(x\right)\right]}}{\log\left(x\right)^{2}}$$
Schritt 5 — Konstantenregel
📖 Regel
$$\frac{d}{dx}(c)=0$$
Mit:
- $c = 1$
🧮 Aktueller Ausdruck
$$\frac{\frac{1}{x^{2} + 1} \cdot \left(2\,x + \bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[1\right]}\,}}}\right)\,\log\left(x\right) - \log\left(x^{2} + 1\right)\,\frac{d}{dx}{\left[\log\left(x\right)\right]}}{\log\left(x\right)^{2}}$$
✨ Nach Konstantenregel
$$\frac{\frac{1}{x^{2} + 1} \cdot \left(2\,x + \bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,0\,}}}\right)\,\log\left(x\right) - \log\left(x^{2} + 1\right)\,\frac{d}{dx}{\left[\log\left(x\right)\right]}}{\log\left(x\right)^{2}}$$
🧹 Vereinfacht
$$\frac{\bbox[lightgreen, padding:0.12em 0.22em]{\vphantom{\dfrac{d}{dx}}\,2\,x \cdot \frac{1}{x^{2} + 1}\,}\,\log\left(x\right) - \log\left(x^{2} + 1\right)\,\frac{d}{dx}{\left[\log\left(x\right)\right]}}{\log\left(x\right)^{2}}$$
Schritt 6 — Ableitung des Logarithmus
📖 Regel
$$\frac{d}{dx}(\log(x)) = \frac{1}{x}$$
🧮 Aktueller Ausdruck
$$\frac{2\,x \cdot \frac{1}{x^{2} + 1}\,\log\left(x\right) - \log\left(x^{2} + 1\right) \cdot \bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\log\left(x\right)\right]}\,}}}}{\log\left(x\right)^{2}}$$
✨ Nach Ableitung des Logarithmus
$$\frac{2\,x \cdot \frac{1}{x^{2} + 1}\,\log\left(x\right) - \log\left(x^{2} + 1\right) \cdot \bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{x}\,}}}}{\log\left(x\right)^{2}}$$
Endergebnis
$$\frac{\frac{2\,x \cdot 1}{x^{2} + 1}\,\log\left(x\right) - \frac{\log\left(x^{2} + 1\right) \cdot 1}{x}}{\log\left(x\right)^{2}}$$
Direkt berechnet (Maxima)
Abgleich des Ergebnisses mit dem Computeralgebrasystem Maxima:
$$\frac{2\,x}{\left(x^{2} + 1\right)\,\log\left(x\right)} - \frac{\log\left(x^{2} + 1\right)}{x \cdot \log\left(x\right)^{2}}$$
Abgleich Schrittfolge ↔︎ Maxima: gleich (true)