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