Lösung zur Ableitung der Formel
Eingabefunktion
$$x\log_{2}{\left(x\right)}$$
Start der Ableitung
$$\frac{d}{dx}{\left(\frac{1}{\log\left(2\right)}\,x\log\left(x\right)\right)}$$
Schritt 1 — Produktregel
Regel
$$\frac{d}{dx}(u \cdot v)=u'v+uv'$$
Mit:
- $\frac{1}{\log\left(2\right)}$
- $x$
- $\log\left(x\right)$
Aktueller Ausdruck
$$\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left(\frac{1}{\log\left(2\right)}\,x\log\left(x\right)\right)}\,}}}$$
Nach Produktregel
$$\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left(\frac{1}{\log\left(2\right)}\right)}x\log\left(x\right) + \frac{1}{\log\left(2\right)}\,\frac{d}{dx}{\left(x\log\left(x\right)\right)}\,}}}$$
Schritt 2 — Konstantenregel
Regel
$$\frac{d}{dx}(c)=0$$
Mit:
- $c = \frac{1}{\log\left(2\right)}$
Aktueller Ausdruck
$$\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left(\frac{1}{\log\left(2\right)}\right)}\,}}}x\log\left(x\right) + \frac{1}{\log\left(2\right)}\,\frac{d}{dx}{\left(x\log\left(x\right)\right)}$$
Nach Konstantenregel
$$\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,0\,}}}x\log\left(x\right) + \frac{1}{\log\left(2\right)}\,\frac{d}{dx}{\left(x\log\left(x\right)\right)}$$
Vereinfacht
$$\bbox[lightgreen, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{\log\left(2\right)}\,\frac{d}{dx}{\left(x\log\left(x\right)\right)}\,}}}$$
Schritt 3 — Produktregel
Regel
$$\frac{d}{dx}(u \cdot v)=u'v+uv'$$
Mit:
- $u = x$
- $v = \log\left(x\right)$
Aktueller Ausdruck
$$\frac{1}{\log\left(2\right)}\,\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left(x\log\left(x\right)\right)}\,}}}$$
Nach Produktregel
$$\frac{1}{\log\left(2\right)} \cdot \left(\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left(x\right)}\log\left(x\right) + x\frac{d}{dx}{\left(\log\left(x\right)\right)}\,}}}\right)$$
Schritt 4 — Ableitung der Variablen
Regel
$$\frac{d}{dx}(x)=1$$
Aktueller Ausdruck
$$\frac{1}{\log\left(2\right)} \cdot \left(\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left(x\right)}\,}}}\log\left(x\right) + x\frac{d}{dx}{\left(\log\left(x\right)\right)}\right)$$
Nach Ableitung der Variablen
$$\frac{1}{\log\left(2\right)} \cdot \left(\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,1\,}}}\log\left(x\right) + x\frac{d}{dx}{\left(\log\left(x\right)\right)}\right)$$
Vereinfacht
$$\frac{1}{\log\left(2\right)} \cdot \left(\bbox[lightgreen, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\log\left(x\right)\,}}} + x\frac{d}{dx}{\left(\log\left(x\right)\right)}\right)$$
Schritt 5 — Kettenregel
Regel
$$\frac{d}{dx}f(u)=f'(u)\cdot u'$$
Mit:
- $u = x$
Aktueller Ausdruck
$$\frac{1}{\log\left(2\right)} \cdot \left(\log\left(x\right) + x\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left(\log\left(x\right)\right)}\,}}}\right)$$
Nach Kettenregel
$$\frac{1}{\log\left(2\right)} \cdot \left(\log\left(x\right) + x \cdot \bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{x}\,}}}\right)$$
Ergebnis
$$\frac{1}{\log\left(2\right)} \cdot \left(\log\left(x\right) + x \cdot \frac{1}{x}\right)$$
Direkt berechnet (Maxima)
Ableitung der Eingabefunktion via Maxima:
$$\frac{\log\left(x\right)}{\log\left(2\right)} + \frac{1}{\log\left(2\right)}$$
Abgleich Schrittfolge ↔︎ Maxima:
gleich (true)