derivative of (12x^2-x^4/((4-x^2)^2))

Lösung

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

\frac{8 x ( x ^{2} + 12 )}{( 4 - x ^{2} ) ^{3}}

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

\frac{d}{d x} (12 x^{2} - x^{4}) = 24 x - 4 x^{3}

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

= \frac{( 24 x - 4 x ^{3} ) ( 4 - x ^{2} ) ^{2} - ( - 4 x ( 4 - x ^{2} ) ) ( 12 x ^{2} - x ^{4} )}{( ( 4 - x ^{2} ) ^{2} ) ^{2}}

Vereinfache

\frac{( 24 x - 4 x ^{3} ) ( 4 - x ^{2} ) ^{2} - ( - 4 x ( 4 - x ^{2} ) ) ( 12 x ^{2} - x ^{4} )}{( ( 4 - x ^{2} ) ^{2} ) ^{2}} : \frac{8 x ( x ^{2} + 12 )}{( 4 - x ^{2} ) ^{3}}

= \frac{8 x ( x ^{2} + 12 )}{( 4 - x ^{2} ) ^{3}}

d er i v a t i v e f (x) = (x - 3) (2 x + 4)

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d er i v a t i v e 7 e^{- x}

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\int sin (3 x) + 10 cos (x) d x