derivative of (2xcos(x^2)/(sin(x^2)+1))

Lösung

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

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

Wende die Quotientenregel an:

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

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

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

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

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

Vereinfache

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

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

\int \frac{x}{x ^{2} - 16} d x

(x + y) \frac{d y}{d x} = 1

a re a - x^{2} + 9, - 2 x^{2} + 18

x \to 28.3 lim (f (x))

\int tan^{3} (6 x) sec^{10} (6 x) d x