derivative of (2x^2/(x^2+1))

Lösung

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

\frac{4 x}{( x ^{2} + 1 ) ^{2}}

Schritte zur Lösung

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

Entferne die Konstante:

(a \cdot f)^{'} = a \cdot f^{'}

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

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

2 \cdot \frac{2 x ( x ^{2} + 1 ) - 2 x x ^{2}}{( x ^{2} + 1 ) ^{2}} : \frac{4 x}{( x ^{2} + 1 ) ^{2}}

= \frac{4 x}{( x ^{2} + 1 ) ^{2}}

\frac{d}{d x} (1 + \frac{4}{x})

\int \frac{ln ( 3 x + 1 )}{6 x + 2} d x

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x \to 0 lim (\frac{2 x}{e ^{x} - 1})