derivative of-(24x(x^2+3)/((x^2-3)^4))

Lösung

\frac{d}{d x} (- \frac{24 x ( x ^{2} + 3 )}{( x ^{2} - 3 ) ^{4}})

- \frac{24 ( - 5 x ^{4} - 30 x ^{2} - 9 )}{( x ^{2} - 3 ) ^{5}}

Schritte zur Lösung

\frac{d}{d x} (- \frac{24 x ( x ^{2} + 3 )}{( x ^{2} - 3 ) ^{4}})

Entferne die Konstante:

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

= - 24 \frac{d}{d x} (\frac{x ( x ^{2} + 3 )}{( x ^{2} - 3 ) ^{4}})

Wende die Quotientenregel an:

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

= - 24 \cdot \frac{\frac{d}{d x} ( x ( x ^{2} + 3 ) ) ( x ^{2} - 3 ) ^{4} - \frac{d}{d x} ( ( x ^{2} - 3 ) ^{4} ) x ( x ^{2} + 3 )}{( ( x ^{2} - 3 ) ^{4} ) ^{2}}

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

\frac{d}{d x} ((x^{2} - 3)^{4}) = 8 x (x^{2} - 3)^{3}

= - 24 \cdot \frac{( 3 x ^{2} + 3 ) ( x ^{2} - 3 ) ^{4} - 8 x ( x ^{2} - 3 ) ^{3} x ( x ^{2} + 3 )}{( ( x ^{2} - 3 ) ^{4} ) ^{2}}

Vereinfache

- 24 \cdot \frac{( 3 x ^{2} + 3 ) ( x ^{2} - 3 ) ^{4} - 8 x ( x ^{2} - 3 ) ^{3} x ( x ^{2} + 3 )}{( ( x ^{2} - 3 ) ^{4} ) ^{2}} : - \frac{24 ( - 5 x ^{4} - 30 x ^{2} - 9 )}{( x ^{2} - 3 ) ^{5}}

= - \frac{24 ( - 5 x ^{4} - 30 x ^{2} - 9 )}{( x ^{2} - 3 ) ^{5}}

\int cos (x) (6 + 8 sin^{2} (x)) d x

a re a 2 x^{2}, 8 x^{2}, - 3 x + 5

\int \frac{1}{sin ^{2} ( x ) cos ^{4} ( x )} d x

\frac{d}{d x} (e^{x} sin (x) + cos (x))

x \to - 5 lim ((5 + x)^{2} - 25)