derivative of ((x^7)/(sqrt(x^{14)+1)})

Lösung

\frac{d}{d x} (\frac{( x ^{7} )}{x ^{14} + 1})

\frac{7 x ^{6}}{( x ^{14} + 1 ) x ^{14} + 1}

Schritte zur Lösung

\frac{d}{d x} (\frac{( x ^{7} )}{x ^{14} + 1})

Wende die Quotientenregel an:

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

= \frac{\frac{d}{d x} ( x ^{7} ) x ^{14} + 1 - \frac{d}{d x} ( x ^{14} + 1 ) x ^{7}}{( x ^{14} + 1 ) ^{2}}

\frac{d}{d x} (x^{7}) = 7 x^{6}

\frac{d}{d x} (x^{14} + 1) = \frac{7 x ^{13}}{x ^{14} + 1}

= \frac{7 x ^{6} x ^{14} + 1 - \frac{7 x ^{13}}{x ^{14} + 1} x ^{7}}{( x ^{14} + 1 ) ^{2}}

Vereinfache

\frac{7 x ^{6} x ^{14} + 1 - \frac{7 x ^{13}}{x ^{14} + 1} x ^{7}}{( x ^{14} + 1 ) ^{2}} : \frac{7 x ^{6}}{( x ^{14} + 1 ) x ^{14} + 1}

= \frac{7 x ^{6}}{( x ^{14} + 1 ) x ^{14} + 1}

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