derivative (x^3+2x^2-1)/(x+5)

解

導関数

\frac{x ^{3} + 2 x ^{2} - 1}{x + 5}

\frac{2 x ^{3} + 17 x ^{2} + 20 x + 1}{( x + 5 ) ^{2}}

解答ステップ

\frac{d}{d x} (\frac{x ^{3} + 2 x ^{2} - 1}{x + 5})

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 ^{3} + 2 x ^{2} - 1 ) ( x + 5 ) - \frac{d}{d x} ( x + 5 ) ( x ^{3} + 2 x ^{2} - 1 )}{( x + 5 ) ^{2}}

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

\frac{d}{d x} (x + 5) = 1

= \frac{( 3 x ^{2} + 4 x ) ( x + 5 ) - 1 \cdot ( x ^{3} + 2 x ^{2} - 1 )}{( x + 5 ) ^{2}}

簡素化

\frac{( 3 x ^{2} + 4 x ) ( x + 5 ) - 1 \cdot ( x ^{3} + 2 x ^{2} - 1 )}{( x + 5 ) ^{2}} : \frac{2 x ^{3} + 17 x ^{2} + 20 x + 1}{( x + 5 ) ^{2}}

= \frac{2 x ^{3} + 17 x ^{2} + 20 x + 1}{( x + 5 ) ^{2}}