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