Lösung zur Ableitung der Formel
Eingabefunktion
$$\frac{\sin\left(x\right)}{\left(x + 1\right)^{2}}$$
Start der Ableitung
$$\frac{d}{dx}{\left[\frac{\sin\left(x\right)}{\left(x + 1\right)^{2}}\right]}$$
Schritt 1 — Quotientenregel
📖 Regel
$$\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{u'v-uv'}{v^{2}}$$
Mit:
- $u = \sin\left(x\right)$
- $v = \left(x + 1\right)^{2}$
🧮 Aktueller Ausdruck
$$\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\frac{\sin\left(x\right)}{\left(x + 1\right)^{2}}\right]}\,}}}$$
✨ Nach Quotientenregel
$$\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{\frac{d}{dx}{\left[\sin\left(x\right)\right]} \cdot \left(x + 1\right)^{2} - \sin\left(x\right)\,\frac{d}{dx}{\left[\left(x + 1\right)^{2}\right]}}{\left(\left(x + 1\right)^{2}\right)^{2}}\,}}}$$
🧹 Vereinfacht
$$\frac{\frac{d}{dx}{\left[\sin\left(x\right)\right]} \cdot \left(x + 1\right)^{2} - \sin\left(x\right)\,\frac{d}{dx}{\left[\left(x + 1\right)^{2}\right]}}{\bbox[lightgreen, padding:0.12em 0.22em]{\vphantom{\dfrac{d}{dx}}\,\left(x + 1\right)^{4}\,}}$$
Schritt 2 — Ableitung des Sinus
📖 Regel
$$\frac{d}{dx}(\sin(x)) = \cos(x)$$
🧮 Aktueller Ausdruck
$$\frac{\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\sin\left(x\right)\right]}\,}}} \cdot \left(x + 1\right)^{2} - \sin\left(x\right)\,\frac{d}{dx}{\left[\left(x + 1\right)^{2}\right]}}{\left(x + 1\right)^{4}}$$
✨ Nach Ableitung des Sinus
$$\frac{\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\cos\left(x\right)\,}}} \cdot \left(x + 1\right)^{2} - \sin\left(x\right)\,\frac{d}{dx}{\left[\left(x + 1\right)^{2}\right]}}{\left(x + 1\right)^{4}}$$
🧹 Vereinfacht
$$\frac{\bbox[lightgreen, padding:0.12em 0.22em]{\vphantom{\dfrac{d}{dx}}\,\left(x + 1\right)^{2}\,\cos\left(x\right)\,} - \sin\left(x\right)\,\frac{d}{dx}{\left[\left(x + 1\right)^{2}\right]}}{\left(x + 1\right)^{4}}$$
Schritt 3 — Potenzregel
📖 Regel
$$\frac{d}{dx}\left(u^{c}\right)=c\,u^{c-1}\,u'$$
Mit:
- $u = x + 1$
- $c = 2$
🧮 Aktueller Ausdruck
$$\frac{\left(x + 1\right)^{2}\,\cos\left(x\right) - \sin\left(x\right) \cdot \bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[\left(x + 1\right)^{2}\right]}\,}}}}{\left(x + 1\right)^{4}}$$
✨ Nach Potenzregel
$$\frac{\left(x + 1\right)^{2}\,\cos\left(x\right) - \sin\left(x\right) \cdot \bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,2 \cdot \left(x + 1\right)^{2 - 1}\,\frac{d}{dx}{\left[x + 1\right]}\,}}}}{\left(x + 1\right)^{4}}$$
🧹 Vereinfacht
$$\frac{\left(x + 1\right)^{2}\,\cos\left(x\right) - \sin\left(x\right) \cdot 2 \cdot \left(x + 1\right)^{\bbox[lightgreen, padding:0.12em 0.22em]{\vphantom{\dfrac{d}{dx}}\,1\,}}\,\frac{d}{dx}{\left[x + 1\right]}}{\left(x + 1\right)^{4}}$$
Schritt 4 — Summen-/Differenzenregel
📖 Regel
$$\frac{d}{dx}(u \pm v \pm \dots) = \frac{d}{dx}(u) \pm \frac{d}{dx}(v) \pm \dots$$
Mit:
- $u = x$
- $v = 1$
🧮 Aktueller Ausdruck
$$\frac{\left(x + 1\right)^{2}\,\cos\left(x\right) - \sin\left(x\right) \cdot 2\,\left(x + 1\right) \cdot \bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x + 1\right]}\,}}}}{\left(x + 1\right)^{4}}$$
✨ Nach Summen-/Differenzenregel
$$\frac{\left(x + 1\right)^{2}\,\cos\left(x\right) - \sin\left(x\right) \cdot 2\,\left(x + 1\right) \cdot \bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x\right]} + \frac{d}{dx}{\left[1\right]}\,}}}}{\left(x + 1\right)^{4}}$$
Schritt 5 — Ableitung der Variablen
📖 Regel
$$\frac{d}{dx}(x)=1$$
🧮 Aktueller Ausdruck
$$\frac{\left(x + 1\right)^{2}\,\cos\left(x\right) - \sin\left(x\right) \cdot 2\,\left(x + 1\right) \cdot \left(\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[x\right]}\,}}} + \frac{d}{dx}{\left[1\right]}\right)}{\left(x + 1\right)^{4}}$$
✨ Nach Ableitung der Variablen
$$\frac{\left(x + 1\right)^{2}\,\cos\left(x\right) - \sin\left(x\right) \cdot 2\,\left(x + 1\right) \cdot \left(\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,1\,}}} + \frac{d}{dx}{\left[1\right]}\right)}{\left(x + 1\right)^{4}}$$
Schritt 6 — Konstantenregel
📖 Regel
$$\frac{d}{dx}(c)=0$$
Mit:
- $c = 1$
🧮 Aktueller Ausdruck
$$\frac{\left(x + 1\right)^{2}\,\cos\left(x\right) - \sin\left(x\right) \cdot 2\,\left(x + 1\right) \cdot \left(1 + \bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left[1\right]}\,}}}\right)}{\left(x + 1\right)^{4}}$$
✨ Nach Konstantenregel
$$\frac{\left(x + 1\right)^{2}\,\cos\left(x\right) - \sin\left(x\right) \cdot 2\,\left(x + 1\right) \cdot \left(1 + \bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,0\,}}}\right)}{\left(x + 1\right)^{4}}$$
🧹 Vereinfacht
$$\frac{\left(x + 1\right)^{2}\,\cos\left(x\right) - \bbox[lightgreen, padding:0.12em 0.22em]{\vphantom{\dfrac{d}{dx}}\,2\,\sin\left(x\right)\,\left(x + 1\right)\,}}{\left(x + 1\right)^{4}}$$
Endergebnis
$$-\frac{2\,\sin\left(x\right) - x\,\cos\left(x\right) - \cos\left(x\right)}{\left(x + 1\right)^{3}}$$
Direkt berechnet (Maxima)
Abgleich des Ergebnisses mit dem Computeralgebrasystem Maxima:
$$\frac{\cos\left(x\right)}{\left(x + 1\right)^{2}} - \frac{2\,\sin\left(x\right)}{\left(x + 1\right)^{3}}$$
Abgleich Schrittfolge ↔︎ Maxima: gleich (true)