derivative of (e^{3x}/(x^6))

Lösung

\frac{d}{d x} (\frac{e ^{3 x}}{x ^{6}})

\frac{3 e ^{3 x} ( x - 2 )}{x ^{7}}

Schritte zur Lösung

\frac{d}{d x} (\frac{e ^{3 x}}{x ^{6}})

Wende die Quotientenregel an:

(\frac{f}{g})^{'} = \frac{f ^{'} \cdot g - g ^{'} \cdot f}{g ^{2}}

= \frac{\frac{d}{d x} ( e ^{3 x} ) x ^{6} - \frac{d}{d x} ( x ^{6} ) e ^{3 x}}{( x ^{6} ) ^{2}}

\frac{d}{d x} (e^{3 x}) = e^{3 x} \cdot 3

\frac{d}{d x} (x^{6}) = 6 x^{5}

= \frac{e ^{3 x} \cdot 3 x ^{6} - 6 x ^{5} e ^{3 x}}{( x ^{6} ) ^{2}}

Vereinfache

\frac{e ^{3 x} \cdot 3 x ^{6} - 6 x ^{5} e ^{3 x}}{( x ^{6} ) ^{2}} : \frac{3 e ^{3 x} ( x - 2 )}{x ^{7}}

= \frac{3 e ^{3 x} ( x - 2 )}{x ^{7}}

x \to 0 lim (\frac{4 x}{x})

\frac{d}{d t} (d cos (t) + t^{2} sin (t))

\frac{d}{d x} (r (x) e^{r (x) x})

\frac{d}{d x} (\frac{sec ( x )}{7 + sec ( x )})

\frac{d}{d x} (ln (x + x^{2} - 8))