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