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