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
$$\frac{\cos\left(x\right)\left(\sin\left(x\right) + \cos\left(x\right)\right) - \sin\left(x\right)\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left(\sin\left(x\right) + \cos\left(x\right)\right)}\,}}}}{\left(\sin\left(x\right) + \cos\left(x\right)\right)^{2}}$$
Nach Summen-/Differenzenregel
$$\frac{\cos\left(x\right)\left(\sin\left(x\right) + \cos\left(x\right)\right) - \sin\left(x\right) \cdot \left(\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left(\sin\left(x\right)\right)} + \frac{d}{dx}{\left(\cos\left(x\right)\right)}\,}}}\right)}{\left(\sin\left(x\right) + \cos\left(x\right)\right)^{2}}$$
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
$$\frac{\cos\left(x\right)\left(\sin\left(x\right) + \cos\left(x\right)\right) - \bbox[lightgreen, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\sin\left(x\right) \cdot \left(\cos\left(x\right) - \sin\left(x\right)\right)\,}}}}{\left(\sin\left(x\right) + \cos\left(x\right)\right)^{2}}$$
Ergebnis
$$\frac{\cos\left(x\right)\left(\sin\left(x\right) + \cos\left(x\right)\right) - \sin\left(x\right) \cdot \left(\cos\left(x\right) - \sin\left(x\right)\right)}{\left(\sin\left(x\right) + \cos\left(x\right)\right)^{2}}$$
Direkt berechnet (Maxima)
Ableitung der Eingabefunktion via 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)