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
$$\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{\frac{d}{dx}{\left[x^{3}\right]}\,\sin\left(\log\left(x\right)\right) - x^{3}\,\frac{d}{dx}{\left[\sin\left(\log\left(x\right)\right)\right]}}{\sin\left(\log\left(x\right)\right)^{2}}\,}}}$$
Schritt 2 — Potenzregel (Spezialfall)
📖 Regel
$$\frac{d}{dx}\left(u^{c}\right)=c\,u^{c-1}$$
Mit:
- $u = x$
- $c = 3$
🧮 Aktueller Ausdruck
$$\frac{\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x^{3}\right]}\,}}}\,\sin\left(\log\left(x\right)\right) - x^{3}\,\frac{d}{dx}{\left[\sin\left(\log\left(x\right)\right)\right]}}{\sin\left(\log\left(x\right)\right)^{2}}$$
✨ Nach Potenzregel (Spezialfall)
$$\frac{\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,3\,x^{3 - 1}\,}}}\,\sin\left(\log\left(x\right)\right) - x^{3}\,\frac{d}{dx}{\left[\sin\left(\log\left(x\right)\right)\right]}}{\sin\left(\log\left(x\right)\right)^{2}}$$
🧹 Vereinfacht
$$\frac{3\,x^{\bbox[lightgreen, padding:0.12em 0.22em]{\vphantom{\dfrac{d}{dx}}\,2\,}}\,\sin\left(\log\left(x\right)\right) - x^{3}\,\frac{d}{dx}{\left[\sin\left(\log\left(x\right)\right)\right]}}{\sin\left(\log\left(x\right)\right)^{2}}$$
Schritt 3 — Ableitung des Sinus (Kettenregel)
📖 Regel
$$\frac{d}{dx}(\sin(u)) = \cos(u) \cdot u'$$
Mit:
- $u = \log\left(x\right)$
🧮 Aktueller Ausdruck
$$\frac{3\,x^{2}\,\sin\left(\log\left(x\right)\right) - x^{3} \cdot \bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\sin\left(\log\left(x\right)\right)\right]}\,}}}}{\sin\left(\log\left(x\right)\right)^{2}}$$
✨ Nach Ableitung des Sinus (Kettenregel)
$$\frac{3\,x^{2}\,\sin\left(\log\left(x\right)\right) - x^{3} \cdot \bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\cos\left(\log\left(x\right)\right)\,\frac{d}{dx}{\left[\log\left(x\right)\right]}\,}}}}{\sin\left(\log\left(x\right)\right)^{2}}$$
Schritt 4 — Ableitung des Logarithmus
📖 Regel
$$\frac{d}{dx}(\log(x)) = \frac{1}{x}$$
🧮 Aktueller Ausdruck
$$\frac{3\,x^{2}\,\sin\left(\log\left(x\right)\right) - x^{3} \cdot \cos\left(\log\left(x\right)\right) \cdot \bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\log\left(x\right)\right]}\,}}}}{\sin\left(\log\left(x\right)\right)^{2}}$$
✨ Nach Ableitung des Logarithmus
$$\frac{3\,x^{2}\,\sin\left(\log\left(x\right)\right) - x^{3} \cdot \cos\left(\log\left(x\right)\right) \cdot \bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{x}\,}}}}{\sin\left(\log\left(x\right)\right)^{2}}$$
Endergebnis
$$\frac{3 \cdot \sin\left(\log\left(x\right)\right)\,x^{2} - x^{2}\,\cos\left(\log\left(x\right)\right)}{\sin\left(\log\left(x\right)\right)^{2}}$$
Direkt berechnet (Maxima)
Abgleich des Ergebnisses mit dem Computeralgebrasystem Maxima:
$$\frac{3\,x^{2}}{\sin\left(\log\left(x\right)\right)} - \frac{x^{2}\,\cos\left(\log\left(x\right)\right)}{\sin\left(\log\left(x\right)\right)^{2}}$$
Abgleich Schrittfolge ↔︎ Maxima: gleich (true)