derivative of (-cos(x)/(4x^2+2x-3))

解

\frac{d}{d x} (\frac{- cos ( x )}{4 x ^{2} + 2 x - 3})

- \frac{- sin ( x ) ( 4 x ^{2} + 2 x - 3 ) - cos ( x ) ( 8 x + 2 )}{( 4 x ^{2} + 2 x - 3 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{- cos ( x )}{4 x ^{2} + 2 x - 3})

簡素化

\frac{- cos ( x )}{4 x ^{2} + 2 x - 3} : - \frac{cos ( x )}{4 x ^{2} + 2 x - 3}

= \frac{d}{d x} (- \frac{cos ( x )}{4 x ^{2} + 2 x - 3})

定数を除去:

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

= - \frac{d}{d x} (\frac{cos ( x )}{4 x ^{2} + 2 x - 3})

商の法則を適用:

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

= - \frac{\frac{d}{d x} ( cos ( x ) ) ( 4 x ^{2} + 2 x - 3 ) - \frac{d}{d x} ( 4 x ^{2} + 2 x - 3 ) cos ( x )}{( 4 x ^{2} + 2 x - 3 ) ^{2}}

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

\frac{d}{d x} (4 x^{2} + 2 x - 3) = 8 x + 2

= - \frac{( - sin ( x ) ) ( 4 x ^{2} + 2 x - 3 ) - ( 8 x + 2 ) cos ( x )}{( 4 x ^{2} + 2 x - 3 ) ^{2}}

簡素化

= - \frac{- sin ( x ) ( 4 x ^{2} + 2 x - 3 ) - cos ( x ) ( 8 x + 2 )}{( 4 x ^{2} + 2 x - 3 ) ^{2}}