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

Lösung

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

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

Schritte zur Lösung

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

Entferne die Konstante:

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

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

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

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

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

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

Vereinfache

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

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

f^{^{'}} (x) = x^{3}

ma c l a u r in \frac{1}{x + 5}

\frac{d}{d x} (x + i y (x))

(x + y)^{2} d x + (2 x y + x^{2} - 5) d y = 0, y (1) = 1

\int_{0}^{8} \frac{1}{x ^{2} + 16} d x