derivative of (3cos(x)/(3x^2+3x-1))

解

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

\frac{3 ( - sin ( x ) ( 3 x ^{2} + 3 x - 1 ) - cos ( x ) ( 6 x + 3 ) )}{( 3 x ^{2} + 3 x - 1 ) ^{2}}

解答ステップ

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

定数を除去:

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

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

商の法則を適用:

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

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

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

\frac{d}{d x} (3 x^{2} + 3 x - 1) = 6 x + 3

= 3 \cdot \frac{( - sin ( x ) ) ( 3 x ^{2} + 3 x - 1 ) - ( 6 x + 3 ) cos ( x )}{( 3 x ^{2} + 3 x - 1 ) ^{2}}

簡素化

3 \cdot \frac{( - sin ( x ) ) ( 3 x ^{2} + 3 x - 1 ) - ( 6 x + 3 ) cos ( x )}{( 3 x ^{2} + 3 x - 1 ) ^{2}} : \frac{3 ( - sin ( x ) ( 3 x ^{2} + 3 x - 1 ) - cos ( x ) ( 6 x + 3 ) )}{( 3 x ^{2} + 3 x - 1 ) ^{2}}

= \frac{3 ( - sin ( x ) ( 3 x ^{2} + 3 x - 1 ) - cos ( x ) ( 6 x + 3 ) )}{( 3 x ^{2} + 3 x - 1 ) ^{2}}