derivative of (2x/(sqrt(4x^2+1)))

Lösung

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

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

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

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

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

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

Vereinfache

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

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

\int \frac{cos ( x )}{1 + cos ( x )} d x

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