derivative of (ax^3+bx+c/(dx^3-e))

解

\frac{d}{d x} (\frac{a x ^{3} + b x + c}{d x ^{3} - e})

\frac{- 2 b d x ^{3} - 3 e a x ^{2} - 3 c d x ^{2} - e b}{( d x ^{3} - e ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{a x ^{3} + b x + c}{d x ^{3} - e})

商の法則を適用:

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

= \frac{\frac{d}{d x} ( a x ^{3} + b x + c ) ( d x ^{3} - e ) - \frac{d}{d x} ( d x ^{3} - e ) ( a x ^{3} + b x + c )}{( d x ^{3} - e ) ^{2}}

\frac{d}{d x} (a x^{3} + b x + c) = 3 a x^{2} + b

\frac{d}{d x} (d x^{3} - e) = 3 d x^{2}

= \frac{( 3 a x ^{2} + b ) ( d x ^{3} - e ) - 3 d x ^{2} ( a x ^{3} + b x + c )}{( d x ^{3} - e ) ^{2}}

拡張

(3 a x^{2} + b) (d x^{3} - e) - 3 d x^{2} (a x^{3} + b x + c) : - 2 b d x^{3} - 3 e a x^{2} - 3 c d x^{2} - e b

= \frac{- 2 b d x ^{3} - 3 e a x ^{2} - 3 c d x ^{2} - e b}{( d x ^{3} - e ) ^{2}}