derivative of (6x^2/(2y+cos(y)))

解

\frac{d}{d x} (\frac{6 x ^{2}}{2 y + cos ( y )})

\frac{6 ( 2 x ( 2 y + cos ( y ) ) - x ^{2} ( 2 \frac{d y}{d x} - sin ( y ) \frac{d y}{d x} ) )}{( 2 y + cos ( y ) ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{6 x ^{2}}{2 y + cos ( y )})

定数を除去:

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

= 6 \frac{d}{d x} (\frac{x ^{2}}{2 y + cos ( y )})

商の法則を適用:

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

= 6 \cdot \frac{\frac{d}{d x} ( x ^{2} ) ( 2 y + cos ( y ) ) - \frac{d}{d x} ( 2 y + cos ( y ) ) x ^{2}}{( 2 y + cos ( y ) ) ^{2}}

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

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

= 6 \cdot \frac{2 x ( 2 y + cos ( y ) ) - ( 2 \frac{d y}{d x} - sin ( y ) \frac{d y}{d x} ) x ^{2}}{( 2 y + cos ( y ) ) ^{2}}

分数を乗じる:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{6 ( 2 x ( 2 y + cos ( y ) ) - x ^{2} ( 2 \frac{d y}{d x} - sin ( y ) \frac{d y}{d x} ) )}{( 2 y + cos ( y ) ) ^{2}}