derivative of (x^4-x^3-8/((x-3)(x+2)))

解

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

\frac{2 x ^{5} - 4 x ^{4} - 22 x ^{3} + 18 x ^{2} + 16 x - 8}{( x - 3 ) ^{2} ( x + 2 ) ^{2}}

解答ステップ

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

商の法則を適用:

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

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

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

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

= \frac{( 4 x ^{3} - 3 x ^{2} ) ( x - 3 ) ( x + 2 ) - ( 2 x - 1 ) ( x ^{4} - x ^{3} - 8 )}{( ( x - 3 ) ( x + 2 ) ) ^{2}}

簡素化

\frac{( 4 x ^{3} - 3 x ^{2} ) ( x - 3 ) ( x + 2 ) - ( 2 x - 1 ) ( x ^{4} - x ^{3} - 8 )}{( ( x - 3 ) ( x + 2 ) ) ^{2}} : \frac{2 x ^{5} - 4 x ^{4} - 22 x ^{3} + 18 x ^{2} + 16 x - 8}{( x - 3 ) ^{2} ( x + 2 ) ^{2}}

= \frac{2 x ^{5} - 4 x ^{4} - 22 x ^{3} + 18 x ^{2} + 16 x - 8}{( x - 3 ) ^{2} ( x + 2 ) ^{2}}