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