Lösung zur Ableitung der Formel
Eingabefunktion
$$\frac{\log_{10}{\left(x\right)}}{x}$$
Start der Ableitung
$$\frac{d}{dx}{\left(\frac{\frac{1}{\log\left(10\right)}\,\log\left(x\right)}{x}\right)}$$
Schritt 1 — Quotientenregel
Regel
$$\frac{d}{dx}\left(\frac{u}{v}\right)=\frac{u'v-uv'}{v^{2}}$$
Mit:
- $u = \frac{1}{\log\left(10\right)}\,\log\left(x\right)$
- $v = x$
Aktueller Ausdruck
$$\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left(\frac{\frac{1}{\log\left(10\right)}\,\log\left(x\right)}{x}\right)}\,}}}$$
Nach Quotientenregel
$$\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{\frac{d}{dx}{\left(\frac{1}{\log\left(10\right)}\,\log\left(x\right)\right)}x - \frac{1}{\log\left(10\right)}\,\log\left(x\right)\frac{d}{dx}{\left(x\right)}}{x^{2}}\,}}}$$
Schritt 2 — Faktorregel
Regel
$$\frac{d}{dx}(c\,u)=c\,u'$$
Mit:
- $c = \frac{1}{\log\left(10\right)}$
- $u = \log\left(x\right)$
Aktueller Ausdruck
$$\frac{\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left(\frac{1}{\log\left(10\right)}\,\log\left(x\right)\right)}\,}}}x - \frac{1}{\log\left(10\right)}\,\log\left(x\right)\frac{d}{dx}{\left(x\right)}}{x^{2}}$$
Nach Faktorregel
$$\frac{\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{\log\left(10\right)}\,\frac{d}{dx}{\left(\log\left(x\right)\right)}\,}}}x - \frac{1}{\log\left(10\right)}\,\log\left(x\right)\frac{d}{dx}{\left(x\right)}}{x^{2}}$$
Schritt 3 — Kettenregel
Regel
$$\frac{d}{dx}f(u)=f'(u)\cdot u'$$
Mit:
- $u = x$
Aktueller Ausdruck
$$\frac{\frac{1}{\log\left(10\right)}\,\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left(\log\left(x\right)\right)}\,}}}x - \frac{1}{\log\left(10\right)}\,\log\left(x\right)\frac{d}{dx}{\left(x\right)}}{x^{2}}$$
Nach Kettenregel
$$\frac{\frac{1}{\log\left(10\right)} \cdot \bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{x}\,}}}x - \frac{1}{\log\left(10\right)}\,\log\left(x\right)\frac{d}{dx}{\left(x\right)}}{x^{2}}$$
Schritt 4 — Ableitung der Variablen
Regel
$$\frac{d}{dx}(x)=1$$
Aktueller Ausdruck
$$\frac{\frac{1}{\log\left(10\right)} \cdot \frac{1}{x}x - \frac{1}{\log\left(10\right)}\,\log\left(x\right)\bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{d}{dx}{\left(x\right)}\,}}}}{x^{2}}$$
Nach Ableitung der Variablen
$$\frac{\frac{1}{\log\left(10\right)} \cdot \frac{1}{x}x - \frac{1}{\log\left(10\right)}\,\log\left(x\right) \cdot \bbox[yellow, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,1\,}}}}{x^{2}}$$
Vereinfacht
$$\frac{\frac{1}{\log\left(10\right)} \cdot \frac{1}{x}x - \bbox[lightgreen, padding:0.12em 0.22em]{{\color{black}{\vphantom{\dfrac{d}{dx}}\,\frac{1}{\log\left(10\right)}\,\log\left(x\right)\,}}}}{x^{2}}$$
Ergebnis
$$\frac{\frac{1}{\log\left(10\right)} \cdot \frac{1}{x}x - \frac{1}{\log\left(10\right)}\,\log\left(x\right)}{x^{2}}$$
Direkt berechnet (Maxima)
Ableitung der Eingabefunktion via Maxima:
$$\frac{1}{\log\left(10\right)} \cdot \frac{1}{x^{2}} - \frac{\log\left(x\right)}{\log\left(10\right)x^{2}}$$
Abgleich Schrittfolge ↔︎ Maxima:
gleich (true)