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