derivative of (4x/(5x^2+3))

Lösung

\frac{d}{d x} (\frac{4 x}{5 x ^{2} + 3})

\frac{4 ( - 5 x ^{2} + 3 )}{( 5 x ^{2} + 3 ) ^{2}}

Schritte zur Lösung

\frac{d}{d x} (\frac{4 x}{5 x ^{2} + 3})

Entferne die Konstante:

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

= 4 \frac{d}{d x} (\frac{x}{5 x ^{2} + 3})

Wende die Quotientenregel an:

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

= 4 \cdot \frac{\frac{d x}{d x} ( 5 x ^{2} + 3 ) - \frac{d}{d x} ( 5 x ^{2} + 3 ) x}{( 5 x ^{2} + 3 ) ^{2}}

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

\frac{d}{d x} (5 x^{2} + 3) = 10 x

= 4 \cdot \frac{1 \cdot ( 5 x ^{2} + 3 ) - 10 xx}{( 5 x ^{2} + 3 ) ^{2}}

Vereinfache

4 \cdot \frac{1 \cdot ( 5 x ^{2} + 3 ) - 10 xx}{( 5 x ^{2} + 3 ) ^{2}} : \frac{4 ( - 5 x ^{2} + 3 )}{( 5 x ^{2} + 3 ) ^{2}}

= \frac{4 ( - 5 x ^{2} + 3 )}{( 5 x ^{2} + 3 ) ^{2}}