derivative y=(sqrt(x)+x^3)/(x^2)

Lösung

ableitung von

y = \frac{x + x ^{3}}{x ^{2}}

\frac{2 x ^{3} - 3 x}{2 x ^{3}}

Schritte zur Lösung

\frac{d}{d x} (\frac{x + x ^{3}}{x ^{2}})

Wende die Quotientenregel an:

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

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

\frac{d}{d x} (x + x^{3}) = \frac{1}{2 x} + 3 x^{2}

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

= \frac{( \frac{1}{2 x} + 3 x ^{2} ) x ^{2} - 2 x ( x + x ^{3} )}{( x ^{2} ) ^{2}}

Vereinfache

\frac{( \frac{1}{2 x} + 3 x ^{2} ) x ^{2} - 2 x ( x + x ^{3} )}{( x ^{2} ) ^{2}} : \frac{2 x ^{3} - 3 x}{2 x ^{3}}

= \frac{2 x ^{3} - 3 x}{2 x ^{3}}

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