derivative of (2x+cos(2x)/(x^2+sin(2x)))

Lösung

\frac{d}{d x} (\frac{2 x + cos ( 2 x )}{x ^{2} + sin ( 2 x )})

- \frac{2 ( - sin ( 2 x ) + x ^{2} + x ^{2} sin ( 2 x ) + 3 x cos ( 2 x ) + 1 )}{( x ^{2} + sin ( 2 x ) ) ^{2}}

Schritte zur Lösung

\frac{d}{d x} (\frac{2 x + cos ( 2 x )}{x ^{2} + sin ( 2 x )})

Wende die Quotientenregel an:

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

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

\frac{d}{d x} (2 x + cos (2 x)) = 2 - sin (2 x) \cdot 2

\frac{d}{d x} (x^{2} + sin (2 x)) = 2 x + cos (2 x) \cdot 2

= \frac{( 2 - sin ( 2 x ) \cdot 2 ) ( x ^{2} + sin ( 2 x ) ) - ( 2 x + cos ( 2 x ) \cdot 2 ) ( 2 x + cos ( 2 x ) )}{( x ^{2} + sin ( 2 x ) ) ^{2}}

Vereinfache

\frac{( 2 - sin ( 2 x ) \cdot 2 ) ( x ^{2} + sin ( 2 x ) ) - ( 2 x + cos ( 2 x ) \cdot 2 ) ( 2 x + cos ( 2 x ) )}{( x ^{2} + sin ( 2 x ) ) ^{2}} : - \frac{2 ( - sin ( 2 x ) + x ^{2} + x ^{2} sin ( 2 x ) + 3 x cos ( 2 x ) + 1 )}{( x ^{2} + sin ( 2 x ) ) ^{2}}

= - \frac{2 ( - sin ( 2 x ) + x ^{2} + x ^{2} sin ( 2 x ) + 3 x cos ( 2 x ) + 1 )}{( x ^{2} + sin ( 2 x ) ) ^{2}}

\frac{\partial}{\partial x} ((x^{2} + y^{2}) e^{- x})

x \to 1 lim (\frac{4}{x ^{2} - 1})

\frac{d}{d x} (cos (y) \frac{d y}{d x})

d er i v a t i v e 5 x^{- 4}

\int (x + 1) \cdot (x - x + 1) d x