derivative of (e^{xx}-e^x/(x^2))

Lösung

\frac{d}{d x} (\frac{e ^{xx} - e ^{x}}{x ^{2}})

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

Schritte zur Lösung

\frac{d}{d x} (\frac{e ^{xx} - e ^{x}}{x ^{2}})

Wende die Quotientenregel an:

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

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

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

\frac{d}{d x} (x^{2}) = 2 x

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

Vereinfache

\frac{( 2 e ^{x^{2}} x - e ^{x} ) x ^{2} - 2 x ( e ^{xx} - e ^{x} )}{( x ^{2} ) ^{2}} : \frac{x ( 2 e ^{x^{2}} x - e ^{x} ) - 2 ( - e ^{x} + e ^{x^{2}} )}{x ^{3}}

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

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