derivative (5-x^2)/((5+x^2)^2)

Lösung

ableitung von

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

- \frac{2 x ( - x ^{2} + 15 )}{( x ^{2} + 5 ) ^{3}}

Schritte zur Lösung

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

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

\frac{( - 2 x ) ( 5 + x ^{2} ) ^{2} - 4 x ( 5 + x ^{2} ) ( 5 - x ^{2} )}{( ( 5 + x ^{2} ) ^{2} ) ^{2}} : - \frac{2 x ( - x ^{2} + 15 )}{( x ^{2} + 5 ) ^{3}}

= - \frac{2 x ( - x ^{2} + 15 )}{( x ^{2} + 5 ) ^{3}}

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

\int \frac{36}{x ^{3}} d x

x \to 1 lim (9)

\int sin (5 x) cos (7 x) d x

a re a \frac{5}{x}, \frac{5}{x ^{2}}, 1, 4