derivative of (4x(2x^2+3)/((1-2x^2)^3))

Lösung

\frac{d}{d x} (\frac{4 x ( 2 x ^{2} + 3 )}{( 1 - 2 x ^{2} ) ^{3}})

\frac{12 ( 4 x ^{4} + 12 x ^{2} + 1 )}{( 1 - 2 x ^{2} ) ^{4}}

Schritte zur Lösung

\frac{d}{d x} (\frac{4 x ( 2 x ^{2} + 3 )}{( 1 - 2 x ^{2} ) ^{3}})

Entferne die Konstante:

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

= 4 \frac{d}{d x} (\frac{x ( 2 x ^{2} + 3 )}{( 1 - 2 x ^{2} ) ^{3}})

Wende die Quotientenregel an:

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

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

\frac{d}{d x} (x (2 x^{2} + 3)) = 6 x^{2} + 3

\frac{d}{d x} ((1 - 2 x^{2})^{3}) = - 12 x (1 - 2 x^{2})^{2}

= 4 \cdot \frac{( 6 x ^{2} + 3 ) ( 1 - 2 x ^{2} ) ^{3} - ( - 12 x ( 1 - 2 x ^{2} ) ^{2} ) x ( 2 x ^{2} + 3 )}{( ( 1 - 2 x ^{2} ) ^{3} ) ^{2}}

Vereinfache

4 \cdot \frac{( 6 x ^{2} + 3 ) ( 1 - 2 x ^{2} ) ^{3} - ( - 12 x ( 1 - 2 x ^{2} ) ^{2} ) x ( 2 x ^{2} + 3 )}{( ( 1 - 2 x ^{2} ) ^{3} ) ^{2}} : \frac{12 ( 4 x ^{4} + 12 x ^{2} + 1 )}{( 1 - 2 x ^{2} ) ^{4}}

= \frac{12 ( 4 x ^{4} + 12 x ^{2} + 1 )}{( 1 - 2 x ^{2} ) ^{4}}

\int \frac{3 x + 2}{( x + 5 ) ( x - 2 )} d x

s \to - 8 lim (4 (s) - 2)

\frac{d}{d x} (\frac{2}{x ^{2}} - 5 x)

\frac{d}{d x} (- \frac{9}{( 6 x + 3 ) ^{\frac{3}{2}}})

d er i v a t i v e g (x) = x^{2} (1 - 2 x)