derivative y=(2x)/(4+x^2)

Lösung

ableitung von

y = \frac{2 x}{4 + x ^{2}}

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende die Quotientenregel an:

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

= 2 \cdot \frac{\frac{d x}{d x} ( 4 + x ^{2} ) - \frac{d}{d x} ( 4 + x ^{2} ) x}{( 4 + x ^{2} ) ^{2}}

\frac{d x}{d x} = 1

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

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

Vereinfache

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

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

\frac{d x}{d t} = \frac{2}{x}

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