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

解

\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}}

解答ステップ

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

定数を除去:

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

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

商の法則を適用:

(\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}}

簡素化

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}}