Lösung zur Ableitung der Formel
Eingabefunktion
$$\sin\left(x\right)^{\cos\left(x\right)} - x$$
Start der Ableitung
$$\frac{d}{dx}{\left[\sin\left(x\right)^{\cos\left(x\right)} - x\right]}$$
Schritt 1 — Summen-/Differenzenregel
📖 Regel
$$\frac{d}{dx}(u \pm v \pm \dots) = \frac{d}{dx}(u) \pm \frac{d}{dx}(v) \pm \dots$$
Mit:
- $u = \sin\left(x\right)^{\cos\left(x\right)}$
- $v = x$
🧮 Aktueller Ausdruck
$$\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\sin\left(x\right)^{\cos\left(x\right)} - x\right]}\,}}}$$
✨ Nach Summen-/Differenzenregel
$$\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\sin\left(x\right)^{\cos\left(x\right)}\right]} - \frac{d}{dx}{\left[x\right]}\,}}}$$
Schritt 2 — Allgemeine Potenzregel
📖 Regel
$$\frac{d}{dx}\left(u^{v}\right)=u^{v}\,\big(v'\,\ln(u)+\frac{v\,u'}{u}\big)$$
Mit:
- $u = \sin\left(x\right)$
- $v = \cos\left(x\right)$
🧮 Aktueller Ausdruck
$$\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\sin\left(x\right)^{\cos\left(x\right)}\right]}\,}}} - \frac{d}{dx}{\left[x\right]}$$
✨ Nach Allgemeine Potenzregel
$$\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\sin\left(x\right)^{\cos\left(x\right)} \cdot \left(\frac{d}{dx}{\left[\cos\left(x\right)\right]}\,\ln\left(\sin\left(x\right)\right) + \cos\left(x\right) \cdot \frac{\frac{d}{dx}{\left[\sin\left(x\right)\right]}}{\sin\left(x\right)}\right)\,}}} - \frac{d}{dx}{\left[x\right]}$$
Schritt 3 — Ableitung des Kosinus
📖 Regel
$$\frac{d}{dx}(\cos(x)) = -\sin(x)$$
🧮 Aktueller Ausdruck
$$\sin\left(x\right)^{\cos\left(x\right)} \cdot \left(\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\cos\left(x\right)\right]}\,}}}\,\ln\left(\sin\left(x\right)\right) + \cos\left(x\right) \cdot \frac{\frac{d}{dx}{\left[\sin\left(x\right)\right]}}{\sin\left(x\right)}\right) - \frac{d}{dx}{\left[x\right]}$$
✨ Nach Ableitung des Kosinus
$$\sin\left(x\right)^{\cos\left(x\right)} \cdot \left(\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,-\sin\left(x\right)\,}}}\,\ln\left(\sin\left(x\right)\right) + \cos\left(x\right) \cdot \frac{\frac{d}{dx}{\left[\sin\left(x\right)\right]}}{\sin\left(x\right)}\right) - \frac{d}{dx}{\left[x\right]}$$
🧹 Vereinfacht
$$\sin\left(x\right)^{\cos\left(x\right)} \cdot \left(\bbox[lightgreen, padding:0.12em 0.22em]{\vphantom{\dfrac{d}{dx}}\,-\sin\left(x\right)\,\ln\left(\sin\left(x\right)\right)\,} + \cos\left(x\right) \cdot \frac{\frac{d}{dx}{\left[\sin\left(x\right)\right]}}{\sin\left(x\right)}\right) - \frac{d}{dx}{\left[x\right]}$$
Schritt 4 — Ableitung des Sinus
📖 Regel
$$\frac{d}{dx}(\sin(x)) = \cos(x)$$
🧮 Aktueller Ausdruck
$$\sin\left(x\right)^{\cos\left(x\right)} \cdot \left(-\sin\left(x\right)\,\ln\left(\sin\left(x\right)\right) + \cos\left(x\right) \cdot \frac{\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\sin\left(x\right)\right]}\,}}}}{\sin\left(x\right)}\right) - \frac{d}{dx}{\left[x\right]}$$
✨ Nach Ableitung des Sinus
$$\sin\left(x\right)^{\cos\left(x\right)} \cdot \left(-\sin\left(x\right)\,\ln\left(\sin\left(x\right)\right) + \cos\left(x\right) \cdot \frac{\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\cos\left(x\right)\,}}}}{\sin\left(x\right)}\right) - \frac{d}{dx}{\left[x\right]}$$
Schritt 5 — Ableitung der Variablen
📖 Regel
$$\frac{d}{dx}(x)=1$$
🧮 Aktueller Ausdruck
$$\sin\left(x\right)^{\cos\left(x\right)} \cdot \left(-\sin\left(x\right)\,\ln\left(\sin\left(x\right)\right) + \cos\left(x\right) \cdot \frac{\cos\left(x\right)}{\sin\left(x\right)}\right) - \bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x\right]}\,}}}$$
✨ Nach Ableitung der Variablen
$$\sin\left(x\right)^{\cos\left(x\right)} \cdot \left(-\sin\left(x\right)\,\ln\left(\sin\left(x\right)\right) + \cos\left(x\right) \cdot \frac{\cos\left(x\right)}{\sin\left(x\right)}\right) - \bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,1\,}}}$$
Endergebnis
$$\sin\left(x\right)^{\cos\left(x\right)}\,\left(\frac{\cos\left(x\right)^{2}}{\sin\left(x\right)} - \sin\left(x\right)\,\log\left(\sin\left(x\right)\right)\right) - 1$$
Direkt berechnet (Maxima)
Abgleich des Ergebnisses mit dem Computeralgebrasystem Maxima:
$$\sin\left(x\right)^{\cos\left(x\right)}\,\left(\frac{\cos\left(x\right)^{2}}{\sin\left(x\right)} - \sin\left(x\right)\,\log\left(\sin\left(x\right)\right)\right) - 1$$
Abgleich Schrittfolge ↔︎ Maxima: gleich (true)